Sums and products of quadratic matrices Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université de Versailles-Saint-Quentin-en-Yvelines)

UCD, October 26, 2017

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Framework F : arbitrary field; χ(F) its characteristic; F an algebraic closure. Let A an F-algebra (associative, with unity 1A ). a ∈ A is quadratic whenever ∃(λ, µ) ∈ F2 ,

a2 = λa + µ1A

i.e. ∃p ∈ F[t] : deg(p) = 2

and p(a) = 0.

Examples: idempotents: a2 = a involutions: a2 = 1A square-zero elements: a2 = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Framework F : arbitrary field; χ(F) its characteristic; F an algebraic closure. Let A an F-algebra (associative, with unity 1A ). a ∈ A is quadratic whenever ∃(λ, µ) ∈ F2 ,

a2 = λa + µ1A

i.e. ∃p ∈ F[t] : deg(p) = 2

and p(a) = 0.

Examples: idempotents: a2 = a involutions: a2 = 1A square-zero elements: a2 = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Framework F : arbitrary field; χ(F) its characteristic; F an algebraic closure. Let A an F-algebra (associative, with unity 1A ). a ∈ A is quadratic whenever ∃(λ, µ) ∈ F2 ,

a2 = λa + µ1A

i.e. ∃p ∈ F[t] : deg(p) = 2

and p(a) = 0.

Examples: idempotents: a2 = a involutions: a2 = 1A square-zero elements: a2 = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Framework F : arbitrary field; χ(F) its characteristic; F an algebraic closure. Let A an F-algebra (associative, with unity 1A ). a ∈ A is quadratic whenever ∃(λ, µ) ∈ F2 ,

a2 = λa + µ1A

i.e. ∃p ∈ F[t] : deg(p) = 2

and p(a) = 0.

Examples: idempotents: a2 = a involutions: a2 = 1A square-zero elements: a2 = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Framework F : arbitrary field; χ(F) its characteristic; F an algebraic closure. Let A an F-algebra (associative, with unity 1A ). a ∈ A is quadratic whenever ∃(λ, µ) ∈ F2 ,

a2 = λa + µ1A

i.e. ∃p ∈ F[t] : deg(p) = 2

and p(a) = 0.

Examples: idempotents: a2 = a involutions: a2 = 1A square-zero elements: a2 = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Framework F : arbitrary field; χ(F) its characteristic; F an algebraic closure. Let A an F-algebra (associative, with unity 1A ). a ∈ A is quadratic whenever ∃(λ, µ) ∈ F2 ,

a2 = λa + µ1A

i.e. ∃p ∈ F[t] : deg(p) = 2

and p(a) = 0.

Examples: idempotents: a2 = a involutions: a2 = 1A square-zero elements: a2 = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Framework F : arbitrary field; χ(F) its characteristic; F an algebraic closure. Let A an F-algebra (associative, with unity 1A ). a ∈ A is quadratic whenever ∃(λ, µ) ∈ F2 ,

a2 = λa + µ1A

i.e. ∃p ∈ F[t] : deg(p) = 2

and p(a) = 0.

Examples: idempotents: a2 = a involutions: a2 = 1A square-zero elements: a2 = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Framework F : arbitrary field; χ(F) its characteristic; F an algebraic closure. Let A an F-algebra (associative, with unity 1A ). a ∈ A is quadratic whenever ∃(λ, µ) ∈ F2 ,

a2 = λa + µ1A

i.e. ∃p ∈ F[t] : deg(p) = 2

and p(a) = 0.

Examples: idempotents: a2 = a involutions: a2 = 1A square-zero elements: a2 = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Famous historical problems: Products of idempotents Theorem (J. Erdos (1967)) M ∈ Mn (F) is a product of idempotents iff M singular or M = In . Theorem (Ballantine (1967)) M ∈ Mn (F) is the product of p idempotents iff rk(M − In ) ≤ p dim(Ker M). Proof of necessity: if M = P1 · · · Pp with P1 , . . . , Pp idempotents then X Im(M − In ) ⊂ Im(Pi − In ) i

whence rk(M−In ) ≤

X i

dim Im(Pi −In ) =

X

dim(Ker Pi ) ≤ p dim(Ker M).

i

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Famous historical problems: Products of idempotents Theorem (J. Erdos (1967)) M ∈ Mn (F) is a product of idempotents iff M singular or M = In . Theorem (Ballantine (1967)) M ∈ Mn (F) is the product of p idempotents iff rk(M − In ) ≤ p dim(Ker M). Proof of necessity: if M = P1 · · · Pp with P1 , . . . , Pp idempotents then X Im(M − In ) ⊂ Im(Pi − In ) i

whence rk(M−In ) ≤

X i

dim Im(Pi −In ) =

X

dim(Ker Pi ) ≤ p dim(Ker M).

i

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Famous historical problems: Products of idempotents Theorem (J. Erdos (1967)) M ∈ Mn (F) is a product of idempotents iff M singular or M = In . Theorem (Ballantine (1967)) M ∈ Mn (F) is the product of p idempotents iff rk(M − In ) ≤ p dim(Ker M). Proof of necessity: if M = P1 · · · Pp with P1 , . . . , Pp idempotents then X Im(M − In ) ⊂ Im(Pi − In ) i

whence rk(M−In ) ≤

X i

dim Im(Pi −In ) =

X

dim(Ker Pi ) ≤ p dim(Ker M).

i

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Famous historical problems: Products of idempotents Theorem (J. Erdos (1967)) M ∈ Mn (F) is a product of idempotents iff M singular or M = In . Theorem (Ballantine (1967)) M ∈ Mn (F) is the product of p idempotents iff rk(M − In ) ≤ p dim(Ker M). Proof of necessity: if M = P1 · · · Pp with P1 , . . . , Pp idempotents then X Im(M − In ) ⊂ Im(Pi − In ) i

whence rk(M−In ) ≤

X i

dim Im(Pi −In ) =

X

dim(Ker Pi ) ≤ p dim(Ker M).

i

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Products of involutions

Theorem (Folklore) M ∈ Mn (F) is a product of involutions iff det M = ±1. Theorem (Djokovi´c, Hoffman-Paige, Wonenburger (circa 1966)) M ∈ GLn (F) is the product of two involutions iff M ≃ M −1 . Proof of necessity: if M = AB where A2 = B 2 = In , then M −1 = B −1 A−1 = BA = BMB −1 .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Products of involutions

Theorem (Folklore) M ∈ Mn (F) is a product of involutions iff det M = ±1. Theorem (Djokovi´c, Hoffman-Paige, Wonenburger (circa 1966)) M ∈ GLn (F) is the product of two involutions iff M ≃ M −1 . Proof of necessity: if M = AB where A2 = B 2 = In , then M −1 = B −1 A−1 = BA = BMB −1 .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Products of involutions

Theorem (Folklore) M ∈ Mn (F) is a product of involutions iff det M = ±1. Theorem (Djokovi´c, Hoffman-Paige, Wonenburger (circa 1966)) M ∈ GLn (F) is the product of two involutions iff M ≃ M −1 . Proof of necessity: if M = AB where A2 = B 2 = In , then M −1 = B −1 A−1 = BA = BMB −1 .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Sums of idempotents

Theorem (Hartwig-Putcha, Wu (1990)) M ∈ Mn (F) is a sum of idempotents iff tr M = k.1F for some integer k ≥ rk(M). Proof of necessity: if M = tr M =

Pp

then

X

X tr Pi = ( rk Pi ).1F

X

rk Pi ≥ rk M.

i

and

i=1 Pi ,

i

i

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Sums of idempotents

Theorem (Hartwig-Putcha, Wu (1990)) M ∈ Mn (F) is a sum of idempotents iff tr M = k.1F for some integer k ≥ rk(M). Proof of necessity: if M = tr M =

Pp

then

X

X tr Pi = ( rk Pi ).1F

X

rk Pi ≥ rk M.

i

and

i=1 Pi ,

i

i

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Length problem for sums of idempotents

Open problem: given p ∈ N, which matrices of Mn (F) that are sums of p idempotents? Complete solution unknown if p ≥ 3. Solution known for p = 2 (Hartwig-Putcha, 1990).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Length problem for sums of idempotents

Open problem: given p ∈ N, which matrices of Mn (F) that are sums of p idempotents? Complete solution unknown if p ≥ 3. Solution known for p = 2 (Hartwig-Putcha, 1990).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Length problem for sums of idempotents

Open problem: given p ∈ N, which matrices of Mn (F) that are sums of p idempotents? Complete solution unknown if p ≥ 3. Solution known for p = 2 (Hartwig-Putcha, 1990).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Our problem Let p, q in F[t] with degree 2. x ∈ A is a (p, q)-sum whenever ∃(a, b) ∈ A2 : x = a + b

and p(a) = q(b) = 0.

x ∈ A is a (p, q)-product whenever ∃(a, b) ∈ A2 : x = ab

and p(a) = q(b) = 0.

Note: the set of all (p, q)-sums (resp. (p, q)-products) in A is a union of conjugacy classes. Problem: if A = Mn (F) (or A = End(V ) where V finite-dimensional vector space over F with dimension n), characterize the (p, q)-sums and the (p, q)-products in terms of canonical forms (Jordan or Frobenius canonical form). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Our problem Let p, q in F[t] with degree 2. x ∈ A is a (p, q)-sum whenever ∃(a, b) ∈ A2 : x = a + b

and p(a) = q(b) = 0.

x ∈ A is a (p, q)-product whenever ∃(a, b) ∈ A2 : x = ab

and p(a) = q(b) = 0.

Note: the set of all (p, q)-sums (resp. (p, q)-products) in A is a union of conjugacy classes. Problem: if A = Mn (F) (or A = End(V ) where V finite-dimensional vector space over F with dimension n), characterize the (p, q)-sums and the (p, q)-products in terms of canonical forms (Jordan or Frobenius canonical form). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Our problem Let p, q in F[t] with degree 2. x ∈ A is a (p, q)-sum whenever ∃(a, b) ∈ A2 : x = a + b

and p(a) = q(b) = 0.

x ∈ A is a (p, q)-product whenever ∃(a, b) ∈ A2 : x = ab

and p(a) = q(b) = 0.

Note: the set of all (p, q)-sums (resp. (p, q)-products) in A is a union of conjugacy classes. Problem: if A = Mn (F) (or A = End(V ) where V finite-dimensional vector space over F with dimension n), characterize the (p, q)-sums and the (p, q)-products in terms of canonical forms (Jordan or Frobenius canonical form). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Our problem Let p, q in F[t] with degree 2. x ∈ A is a (p, q)-sum whenever ∃(a, b) ∈ A2 : x = a + b

and p(a) = q(b) = 0.

x ∈ A is a (p, q)-product whenever ∃(a, b) ∈ A2 : x = ab

and p(a) = q(b) = 0.

Note: the set of all (p, q)-sums (resp. (p, q)-products) in A is a union of conjugacy classes. Problem: if A = Mn (F) (or A = End(V ) where V finite-dimensional vector space over F with dimension n), characterize the (p, q)-sums and the (p, q)-products in terms of canonical forms (Jordan or Frobenius canonical form). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Our problem Let p, q in F[t] with degree 2. x ∈ A is a (p, q)-sum whenever ∃(a, b) ∈ A2 : x = a + b

and p(a) = q(b) = 0.

x ∈ A is a (p, q)-product whenever ∃(a, b) ∈ A2 : x = ab

and p(a) = q(b) = 0.

Note: the set of all (p, q)-sums (resp. (p, q)-products) in A is a union of conjugacy classes. Problem: if A = Mn (F) (or A = End(V ) where V finite-dimensional vector space over F with dimension n), characterize the (p, q)-sums and the (p, q)-products in terms of canonical forms (Jordan or Frobenius canonical form). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Some reduction tricks

We can assume that p and q are monic. (p, q)-sums are (q, p)-sums. In Mn (F), (p, q)-products are (q, p)-products (if M = AB with p(A) = q(B) = 0, then M T = B T AT and q(B T ) = p(AT ) = 0; one concludes by noting that M ≃ M T ). Translation trick for sums: let λ ∈ F. Set p1 := p(t − λ). x = a + b with p(a) = q(b) = 0 iff x + λ1A = (a + λ1A ) + B with p1 (a + λ1A ) = q(b) = 0. Multiplication trick for sums: let λ ∈ F∗ . Set p1 := p(t/λ) and q1 := q(t/λ). Then, x = a + b with p(a) = q(b) = 0 iff λx = λa + λb with p1 (λa) = q1 (λb) = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Some reduction tricks

We can assume that p and q are monic. (p, q)-sums are (q, p)-sums. In Mn (F), (p, q)-products are (q, p)-products (if M = AB with p(A) = q(B) = 0, then M T = B T AT and q(B T ) = p(AT ) = 0; one concludes by noting that M ≃ M T ). Translation trick for sums: let λ ∈ F. Set p1 := p(t − λ). x = a + b with p(a) = q(b) = 0 iff x + λ1A = (a + λ1A ) + B with p1 (a + λ1A ) = q(b) = 0. Multiplication trick for sums: let λ ∈ F∗ . Set p1 := p(t/λ) and q1 := q(t/λ). Then, x = a + b with p(a) = q(b) = 0 iff λx = λa + λb with p1 (λa) = q1 (λb) = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Some reduction tricks

We can assume that p and q are monic. (p, q)-sums are (q, p)-sums. In Mn (F), (p, q)-products are (q, p)-products (if M = AB with p(A) = q(B) = 0, then M T = B T AT and q(B T ) = p(AT ) = 0; one concludes by noting that M ≃ M T ). Translation trick for sums: let λ ∈ F. Set p1 := p(t − λ). x = a + b with p(a) = q(b) = 0 iff x + λ1A = (a + λ1A ) + B with p1 (a + λ1A ) = q(b) = 0. Multiplication trick for sums: let λ ∈ F∗ . Set p1 := p(t/λ) and q1 := q(t/λ). Then, x = a + b with p(a) = q(b) = 0 iff λx = λa + λb with p1 (λa) = q1 (λb) = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Some reduction tricks

We can assume that p and q are monic. (p, q)-sums are (q, p)-sums. In Mn (F), (p, q)-products are (q, p)-products (if M = AB with p(A) = q(B) = 0, then M T = B T AT and q(B T ) = p(AT ) = 0; one concludes by noting that M ≃ M T ). Translation trick for sums: let λ ∈ F. Set p1 := p(t − λ). x = a + b with p(a) = q(b) = 0 iff x + λ1A = (a + λ1A ) + B with p1 (a + λ1A ) = q(b) = 0. Multiplication trick for sums: let λ ∈ F∗ . Set p1 := p(t/λ) and q1 := q(t/λ). Then, x = a + b with p(a) = q(b) = 0 iff λx = λa + λb with p1 (λa) = q1 (λb) = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Some reduction tricks

We can assume that p and q are monic. (p, q)-sums are (q, p)-sums. In Mn (F), (p, q)-products are (q, p)-products (if M = AB with p(A) = q(B) = 0, then M T = B T AT and q(B T ) = p(AT ) = 0; one concludes by noting that M ≃ M T ). Translation trick for sums: let λ ∈ F. Set p1 := p(t − λ). x = a + b with p(a) = q(b) = 0 iff x + λ1A = (a + λ1A ) + B with p1 (a + λ1A ) = q(b) = 0. Multiplication trick for sums: let λ ∈ F∗ . Set p1 := p(t/λ) and q1 := q(t/λ). Then, x = a + b with p(a) = q(b) = 0 iff λx = λa + λb with p1 (λa) = q1 (λb) = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Some reduction tricks

We can assume that p and q are monic. (p, q)-sums are (q, p)-sums. In Mn (F), (p, q)-products are (q, p)-products (if M = AB with p(A) = q(B) = 0, then M T = B T AT and q(B T ) = p(AT ) = 0; one concludes by noting that M ≃ M T ). Translation trick for sums: let λ ∈ F. Set p1 := p(t − λ). x = a + b with p(a) = q(b) = 0 iff x + λ1A = (a + λ1A ) + B with p1 (a + λ1A ) = q(b) = 0. Multiplication trick for sums: let λ ∈ F∗ . Set p1 := p(t/λ) and q1 := q(t/λ). Then, x = a + b with p(a) = q(b) = 0 iff λx = λa + λb with p1 (λa) = q1 (λb) = 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Reduced situation for split polynomials If p and q split, only three cases need to be considered: p = t 2 − αt, q = t 2 − t with α nonzero (matrices of the form αP + Q with P, Q idempotents); Hartwig-Putcha (1990) α = ±1 and F = C; de Seguins Pazzis (2010) arbitrary α, arbitrary field. p = q = t 2 (sums of two square-zero matrices); Wang-Wu (1991) F = C; Botha (2012) arbitrary field. p = t 2 − t and q = t 2 (idempotent plus square-zero) Wang (1995) F = C; de Seguins Pazzis (2012) arbitrary field.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Reduced situation for split polynomials If p and q split, only three cases need to be considered: p = t 2 − αt, q = t 2 − t with α nonzero (matrices of the form αP + Q with P, Q idempotents); Hartwig-Putcha (1990) α = ±1 and F = C; de Seguins Pazzis (2010) arbitrary α, arbitrary field. p = q = t 2 (sums of two square-zero matrices); Wang-Wu (1991) F = C; Botha (2012) arbitrary field. p = t 2 − t and q = t 2 (idempotent plus square-zero) Wang (1995) F = C; de Seguins Pazzis (2012) arbitrary field.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Reduced situation for split polynomials If p and q split, only three cases need to be considered: p = t 2 − αt, q = t 2 − t with α nonzero (matrices of the form αP + Q with P, Q idempotents); Hartwig-Putcha (1990) α = ±1 and F = C; de Seguins Pazzis (2010) arbitrary α, arbitrary field. p = q = t 2 (sums of two square-zero matrices); Wang-Wu (1991) F = C; Botha (2012) arbitrary field. p = t 2 − t and q = t 2 (idempotent plus square-zero) Wang (1995) F = C; de Seguins Pazzis (2012) arbitrary field.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Some additional notation

Jordan cell of size n with respect to λ ∈ F:   λ 1 (0)   .   λ ..  ∈ Mn (F). Jn (λ) :=    ..  . 1 (0) λ Direct sum of two square matrices:   A 0 A ⊕ B := . 0 B

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Some additional notation

Jordan cell of size n with respect to λ ∈ F:   λ 1 (0)   .   λ ..  ∈ Mn (F). Jn (λ) :=    ..  . 1 (0) λ Direct sum of two square matrices:   A 0 A ⊕ B := . 0 B

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Statement of some known results Let p, q in F[t] split and monic with degree 2. p(t) = (t − x1 )(t − x2 ) and q(t) = (t − y1 )(t − y2 ). Set σ := x1 + x2 + y1 + y2 = tr(p) + tr(q) and

 Root(p) + Root(q) = xi + yj | (i, j) ∈ {1, 2}2 .

(set of all (p, q)-sums in F!).

Roughly, the (p, q)-sums are the matrices with spectrum σ “symmetric around" · 2

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Statement of some known results Let p, q in F[t] split and monic with degree 2. p(t) = (t − x1 )(t − x2 ) and q(t) = (t − y1 )(t − y2 ). Set σ := x1 + x2 + y1 + y2 = tr(p) + tr(q) and

 Root(p) + Root(q) = xi + yj | (i, j) ∈ {1, 2}2 .

(set of all (p, q)-sums in F!).

Roughly, the (p, q)-sums are the matrices with spectrum σ “symmetric around" · 2

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Theorem Let M ∈ Mn (F). Then M is a (p, q)-sum iff M similar over F to M1 ⊕ · · · ⊕ Mk , each Mi being of one of the following types: Jn (λ) ⊕ Jn (σ − λ), (λ ∈ F such that λ 6= σ − λ, n > 0); Jn+1 (λ) ⊕ Jn (σ − λ), (n ≥ 0, λ ∈ Root(p) + Root(q)); Jn+2 (λ) ⊕ Jn (σ − λ), ( n ≥ 0, λ ∈ Root(p) + Root(q), exactly one of p and q has a double root); J2n (λ), (n > 0, λ ∈ F such that λ = σ − λ). J2n+1 (λ), (n ≥ 0, λ ∈ Root(p) + Root(q) such that λ = σ − λ). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Theorem Let M ∈ Mn (F). Then M is a (p, q)-sum iff M similar over F to M1 ⊕ · · · ⊕ Mk , each Mi being of one of the following types: Jn (λ) ⊕ Jn (σ − λ), (λ ∈ F such that λ 6= σ − λ, n > 0); Jn+1 (λ) ⊕ Jn (σ − λ), (n ≥ 0, λ ∈ Root(p) + Root(q)); Jn+2 (λ) ⊕ Jn (σ − λ), ( n ≥ 0, λ ∈ Root(p) + Root(q), exactly one of p and q has a double root); J2n (λ), (n > 0, λ ∈ F such that λ = σ − λ). J2n+1 (λ), (n ≥ 0, λ ∈ Root(p) + Root(q) such that λ = σ − λ). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Theorem Let M ∈ Mn (F). Then M is a (p, q)-sum iff M similar over F to M1 ⊕ · · · ⊕ Mk , each Mi being of one of the following types: Jn (λ) ⊕ Jn (σ − λ), (λ ∈ F such that λ 6= σ − λ, n > 0); Jn+1 (λ) ⊕ Jn (σ − λ), (n ≥ 0, λ ∈ Root(p) + Root(q)); Jn+2 (λ) ⊕ Jn (σ − λ), ( n ≥ 0, λ ∈ Root(p) + Root(q), exactly one of p and q has a double root); J2n (λ), (n > 0, λ ∈ F such that λ = σ − λ). J2n+1 (λ), (n ≥ 0, λ ∈ Root(p) + Root(q) such that λ = σ − λ). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Theorem Let M ∈ Mn (F). Then M is a (p, q)-sum iff M similar over F to M1 ⊕ · · · ⊕ Mk , each Mi being of one of the following types: Jn (λ) ⊕ Jn (σ − λ), (λ ∈ F such that λ 6= σ − λ, n > 0); Jn+1 (λ) ⊕ Jn (σ − λ), (n ≥ 0, λ ∈ Root(p) + Root(q)); Jn+2 (λ) ⊕ Jn (σ − λ), ( n ≥ 0, λ ∈ Root(p) + Root(q), exactly one of p and q has a double root); J2n (λ), (n > 0, λ ∈ F such that λ = σ − λ). J2n+1 (λ), (n ≥ 0, λ ∈ Root(p) + Root(q) such that λ = σ − λ). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Theorem Let M ∈ Mn (F). Then M is a (p, q)-sum iff M similar over F to M1 ⊕ · · · ⊕ Mk , each Mi being of one of the following types: Jn (λ) ⊕ Jn (σ − λ), (λ ∈ F such that λ 6= σ − λ, n > 0); Jn+1 (λ) ⊕ Jn (σ − λ), (n ≥ 0, λ ∈ Root(p) + Root(q)); Jn+2 (λ) ⊕ Jn (σ − λ), ( n ≥ 0, λ ∈ Root(p) + Root(q), exactly one of p and q has a double root); J2n (λ), (n > 0, λ ∈ F such that λ = σ − λ). J2n+1 (λ), (n ≥ 0, λ ∈ Root(p) + Root(q) such that λ = σ − λ). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Theorem Let M ∈ Mn (F). Then M is a (p, q)-sum iff M similar over F to M1 ⊕ · · · ⊕ Mk , each Mi being of one of the following types: Jn (λ) ⊕ Jn (σ − λ), (λ ∈ F such that λ 6= σ − λ, n > 0); Jn+1 (λ) ⊕ Jn (σ − λ), (n ≥ 0, λ ∈ Root(p) + Root(q)); Jn+2 (λ) ⊕ Jn (σ − λ), ( n ≥ 0, λ ∈ Root(p) + Root(q), exactly one of p and q has a double root); J2n (λ), (n > 0, λ ∈ F such that λ = σ − λ). J2n+1 (λ), (n ≥ 0, λ ∈ Root(p) + Root(q) such that λ = σ − λ). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Decomposability Let V vector space over F , u ∈ End(V ) V = V1 ⊕ V2 with Vi invariant under u (and dim Vi > 0); Assume that u|Vi (p, q)-sum for all i. Then u|Vi = ai + bi with p(ai ) = q(bi ) = 0. Set a ∈ End(V ) and b ∈ End(V ) s.t. a|Vi = ai and b|Vi = bi . p(a) = 0, q(b) = 0, u = a + b, whence u is a (p, q)-sum. In that case u is decomposable. Problem: understand the indecomposable (p, q)-sums. Warning: if u is a (p, q)-sum and V = V1 ⊕ V2 with V1 , V2 invariant under u, u|V1 and u|V2 might not be (p, q)-sums! Example: p = q = t 2 

 1 0 u↔ . 0 −1 Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Decomposability Let V vector space over F , u ∈ End(V ) V = V1 ⊕ V2 with Vi invariant under u (and dim Vi > 0); Assume that u|Vi (p, q)-sum for all i. Then u|Vi = ai + bi with p(ai ) = q(bi ) = 0. Set a ∈ End(V ) and b ∈ End(V ) s.t. a|Vi = ai and b|Vi = bi . p(a) = 0, q(b) = 0, u = a + b, whence u is a (p, q)-sum. In that case u is decomposable. Problem: understand the indecomposable (p, q)-sums. Warning: if u is a (p, q)-sum and V = V1 ⊕ V2 with V1 , V2 invariant under u, u|V1 and u|V2 might not be (p, q)-sums! Example: p = q = t 2 

 1 0 u↔ . 0 −1 Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Decomposability Let V vector space over F , u ∈ End(V ) V = V1 ⊕ V2 with Vi invariant under u (and dim Vi > 0); Assume that u|Vi (p, q)-sum for all i. Then u|Vi = ai + bi with p(ai ) = q(bi ) = 0. Set a ∈ End(V ) and b ∈ End(V ) s.t. a|Vi = ai and b|Vi = bi . p(a) = 0, q(b) = 0, u = a + b, whence u is a (p, q)-sum. In that case u is decomposable. Problem: understand the indecomposable (p, q)-sums. Warning: if u is a (p, q)-sum and V = V1 ⊕ V2 with V1 , V2 invariant under u, u|V1 and u|V2 might not be (p, q)-sums! Example: p = q = t 2 

 1 0 u↔ . 0 −1 Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Decomposability Let V vector space over F , u ∈ End(V ) V = V1 ⊕ V2 with Vi invariant under u (and dim Vi > 0); Assume that u|Vi (p, q)-sum for all i. Then u|Vi = ai + bi with p(ai ) = q(bi ) = 0. Set a ∈ End(V ) and b ∈ End(V ) s.t. a|Vi = ai and b|Vi = bi . p(a) = 0, q(b) = 0, u = a + b, whence u is a (p, q)-sum. In that case u is decomposable. Problem: understand the indecomposable (p, q)-sums. Warning: if u is a (p, q)-sum and V = V1 ⊕ V2 with V1 , V2 invariant under u, u|V1 and u|V2 might not be (p, q)-sums! Example: p = q = t 2 

 1 0 u↔ . 0 −1 Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Decomposability Let V vector space over F , u ∈ End(V ) V = V1 ⊕ V2 with Vi invariant under u (and dim Vi > 0); Assume that u|Vi (p, q)-sum for all i. Then u|Vi = ai + bi with p(ai ) = q(bi ) = 0. Set a ∈ End(V ) and b ∈ End(V ) s.t. a|Vi = ai and b|Vi = bi . p(a) = 0, q(b) = 0, u = a + b, whence u is a (p, q)-sum. In that case u is decomposable. Problem: understand the indecomposable (p, q)-sums. Warning: if u is a (p, q)-sum and V = V1 ⊕ V2 with V1 , V2 invariant under u, u|V1 and u|V2 might not be (p, q)-sums! Example: p = q = t 2 

 1 0 u↔ . 0 −1 Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Decomposability Let V vector space over F , u ∈ End(V ) V = V1 ⊕ V2 with Vi invariant under u (and dim Vi > 0); Assume that u|Vi (p, q)-sum for all i. Then u|Vi = ai + bi with p(ai ) = q(bi ) = 0. Set a ∈ End(V ) and b ∈ End(V ) s.t. a|Vi = ai and b|Vi = bi . p(a) = 0, q(b) = 0, u = a + b, whence u is a (p, q)-sum. In that case u is decomposable. Problem: understand the indecomposable (p, q)-sums. Warning: if u is a (p, q)-sum and V = V1 ⊕ V2 with V1 , V2 invariant under u, u|V1 and u|V2 might not be (p, q)-sums! Example: p = q = t 2 

 1 0 u↔ . 0 −1 Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Decomposability Let V vector space over F , u ∈ End(V ) V = V1 ⊕ V2 with Vi invariant under u (and dim Vi > 0); Assume that u|Vi (p, q)-sum for all i. Then u|Vi = ai + bi with p(ai ) = q(bi ) = 0. Set a ∈ End(V ) and b ∈ End(V ) s.t. a|Vi = ai and b|Vi = bi . p(a) = 0, q(b) = 0, u = a + b, whence u is a (p, q)-sum. In that case u is decomposable. Problem: understand the indecomposable (p, q)-sums. Warning: if u is a (p, q)-sum and V = V1 ⊕ V2 with V1 , V2 invariant under u, u|V1 and u|V2 might not be (p, q)-sums! Example: p = q = t 2 

 1 0 u↔ . 0 −1 Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Regularity Definition Let u ∈ End(V ) (V finite-dimensional vector space over F). u is (p, q)-regular iff no eigenvalue of u in F is a (p, q)-sum u is (p, q)-exceptional iff all eigenvalues of u in F are (p, q)-sums. Theorem In End(V ), any indecomposable (p, q)-sum is either regular or exceptional. Thus, one needs to study only: the indecomposable regular (p, q)-sums; the indecomposable exceptional (p, q)-sums. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Regularity Definition Let u ∈ End(V ) (V finite-dimensional vector space over F). u is (p, q)-regular iff no eigenvalue of u in F is a (p, q)-sum u is (p, q)-exceptional iff all eigenvalues of u in F are (p, q)-sums. Theorem In End(V ), any indecomposable (p, q)-sum is either regular or exceptional. Thus, one needs to study only: the indecomposable regular (p, q)-sums; the indecomposable exceptional (p, q)-sums. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Regularity Definition Let u ∈ End(V ) (V finite-dimensional vector space over F). u is (p, q)-regular iff no eigenvalue of u in F is a (p, q)-sum u is (p, q)-exceptional iff all eigenvalues of u in F are (p, q)-sums. Theorem In End(V ), any indecomposable (p, q)-sum is either regular or exceptional. Thus, one needs to study only: the indecomposable regular (p, q)-sums; the indecomposable exceptional (p, q)-sums. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Regularity Definition Let u ∈ End(V ) (V finite-dimensional vector space over F). u is (p, q)-regular iff no eigenvalue of u in F is a (p, q)-sum u is (p, q)-exceptional iff all eigenvalues of u in F are (p, q)-sums. Theorem In End(V ), any indecomposable (p, q)-sum is either regular or exceptional. Thus, one needs to study only: the indecomposable regular (p, q)-sums; the indecomposable exceptional (p, q)-sums. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Regularity Definition Let u ∈ End(V ) (V finite-dimensional vector space over F). u is (p, q)-regular iff no eigenvalue of u in F is a (p, q)-sum u is (p, q)-exceptional iff all eigenvalues of u in F are (p, q)-sums. Theorem In End(V ), any indecomposable (p, q)-sum is either regular or exceptional. Thus, one needs to study only: the indecomposable regular (p, q)-sums; the indecomposable exceptional (p, q)-sums. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Regularity Definition Let u ∈ End(V ) (V finite-dimensional vector space over F). u is (p, q)-regular iff no eigenvalue of u in F is a (p, q)-sum u is (p, q)-exceptional iff all eigenvalues of u in F are (p, q)-sums. Theorem In End(V ), any indecomposable (p, q)-sum is either regular or exceptional. Thus, one needs to study only: the indecomposable regular (p, q)-sums; the indecomposable exceptional (p, q)-sums. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Regularity Definition Let u ∈ End(V ) (V finite-dimensional vector space over F). u is (p, q)-regular iff no eigenvalue of u in F is a (p, q)-sum u is (p, q)-exceptional iff all eigenvalues of u in F are (p, q)-sums. Theorem In End(V ), any indecomposable (p, q)-sum is either regular or exceptional. Thus, one needs to study only: the indecomposable regular (p, q)-sums; the indecomposable exceptional (p, q)-sums. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The commutation lemma Let a ∈ A and p = t 2 − tr(p) t + p(0) ∈ F[t] such that p(a) = 0. Set a⋆ := tr(p)1A − a (p-conjugate of a) so that aa⋆ = a⋆ a = p(0)1A. Similarly, take q ∈ F[t] monic with degree 2 and b ∈ A s.t. q(b) = 0. Set b ⋆ the q-conjugate of b. ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = a⋆ b + b ⋆ a. Lemma (Commutation lemma) a and b commute with ab ⋆ + ba⋆ .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The commutation lemma Let a ∈ A and p = t 2 − tr(p) t + p(0) ∈ F[t] such that p(a) = 0. Set a⋆ := tr(p)1A − a (p-conjugate of a) so that aa⋆ = a⋆ a = p(0)1A. Similarly, take q ∈ F[t] monic with degree 2 and b ∈ A s.t. q(b) = 0. Set b ⋆ the q-conjugate of b. ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = a⋆ b + b ⋆ a. Lemma (Commutation lemma) a and b commute with ab ⋆ + ba⋆ .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The commutation lemma Let a ∈ A and p = t 2 − tr(p) t + p(0) ∈ F[t] such that p(a) = 0. Set a⋆ := tr(p)1A − a (p-conjugate of a) so that aa⋆ = a⋆ a = p(0)1A. Similarly, take q ∈ F[t] monic with degree 2 and b ∈ A s.t. q(b) = 0. Set b ⋆ the q-conjugate of b. ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = a⋆ b + b ⋆ a. Lemma (Commutation lemma) a and b commute with ab ⋆ + ba⋆ .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The commutation lemma Let a ∈ A and p = t 2 − tr(p) t + p(0) ∈ F[t] such that p(a) = 0. Set a⋆ := tr(p)1A − a (p-conjugate of a) so that aa⋆ = a⋆ a = p(0)1A. Similarly, take q ∈ F[t] monic with degree 2 and b ∈ A s.t. q(b) = 0. Set b ⋆ the q-conjugate of b. ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = a⋆ b + b ⋆ a. Lemma (Commutation lemma) a and b commute with ab ⋆ + ba⋆ .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The commutation lemma Let a ∈ A and p = t 2 − tr(p) t + p(0) ∈ F[t] such that p(a) = 0. Set a⋆ := tr(p)1A − a (p-conjugate of a) so that aa⋆ = a⋆ a = p(0)1A. Similarly, take q ∈ F[t] monic with degree 2 and b ∈ A s.t. q(b) = 0. Set b ⋆ the q-conjugate of b. ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = a⋆ b + b ⋆ a. Lemma (Commutation lemma) a and b commute with ab ⋆ + ba⋆ .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The commutation lemma Let a ∈ A and p = t 2 − tr(p) t + p(0) ∈ F[t] such that p(a) = 0. Set a⋆ := tr(p)1A − a (p-conjugate of a) so that aa⋆ = a⋆ a = p(0)1A. Similarly, take q ∈ F[t] monic with degree 2 and b ∈ A s.t. q(b) = 0. Set b ⋆ the q-conjugate of b. ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = a⋆ b + b ⋆ a. Lemma (Commutation lemma) a and b commute with ab ⋆ + ba⋆ .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Set u := a + b. Then, ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = tr(q)a + tr(p)b − (a + b)2 + a2 + b 2 = σu − u 2 − (p(0) + q(0))1A . Consequence: v := u 2 − σu commutes with a and b. Now, assume that A = End(V ) and split p(t) = (t − x1 )(t − x2 ) and q(t) = (t − y1 )(t − y2 ) (roots in F) and set Fp,q :=

Y (t − xi − yj ) ∈ F[t]. i,j

u regular (resp. exceptional) iff Fp,q (u) invertible (resp. nilpotent). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Set u := a + b. Then, ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = tr(q)a + tr(p)b − (a + b)2 + a2 + b 2 = σu − u 2 − (p(0) + q(0))1A . Consequence: v := u 2 − σu commutes with a and b. Now, assume that A = End(V ) and split p(t) = (t − x1 )(t − x2 ) and q(t) = (t − y1 )(t − y2 ) (roots in F) and set Fp,q :=

Y (t − xi − yj ) ∈ F[t]. i,j

u regular (resp. exceptional) iff Fp,q (u) invertible (resp. nilpotent). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Set u := a + b. Then, ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = tr(q)a + tr(p)b − (a + b)2 + a2 + b 2 = σu − u 2 − (p(0) + q(0))1A . Consequence: v := u 2 − σu commutes with a and b. Now, assume that A = End(V ) and split p(t) = (t − x1 )(t − x2 ) and q(t) = (t − y1 )(t − y2 ) (roots in F) and set Fp,q :=

Y (t − xi − yj ) ∈ F[t]. i,j

u regular (resp. exceptional) iff Fp,q (u) invertible (resp. nilpotent). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Set u := a + b. Then, ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = tr(q)a + tr(p)b − (a + b)2 + a2 + b 2 = σu − u 2 − (p(0) + q(0))1A . Consequence: v := u 2 − σu commutes with a and b. Now, assume that A = End(V ) and split p(t) = (t − x1 )(t − x2 ) and q(t) = (t − y1 )(t − y2 ) (roots in F) and set Fp,q :=

Y (t − xi − yj ) ∈ F[t]. i,j

u regular (resp. exceptional) iff Fp,q (u) invertible (resp. nilpotent). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Set u := a + b. Then, ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = tr(q)a + tr(p)b − (a + b)2 + a2 + b 2 = σu − u 2 − (p(0) + q(0))1A . Consequence: v := u 2 − σu commutes with a and b. Now, assume that A = End(V ) and split p(t) = (t − x1 )(t − x2 ) and q(t) = (t − y1 )(t − y2 ) (roots in F) and set Fp,q :=

Y (t − xi − yj ) ∈ F[t]. i,j

u regular (resp. exceptional) iff Fp,q (u) invertible (resp. nilpotent). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Set u := a + b. Then, ab ⋆ + ba⋆ = tr(q)a + tr(p)b − (ab + ba) = tr(q)a + tr(p)b − (a + b)2 + a2 + b 2 = σu − u 2 − (p(0) + q(0))1A . Consequence: v := u 2 − σu commutes with a and b. Now, assume that A = End(V ) and split p(t) = (t − x1 )(t − x2 ) and q(t) = (t − y1 )(t − y2 ) (roots in F) and set Fp,q :=

Y (t − xi − yj ) ∈ F[t]. i,j

u regular (resp. exceptional) iff Fp,q (u) invertible (resp. nilpotent). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices



Fp,q = t 2 − σt + (x1 + y1 )(x2 + y2 ) t 2 − σt + (x1 + y2 )(x2 + y1 ) so

Fp,q (u) = Λ(v) for

Λ := t + (x1 + y1 )(x2 + y2 ) t + (x1 + y2 )(x2 + y1 ) ∈ F[t]. For large enough N V = Ker Fp,q (u)N ⊕ Im Fp,q (u)N . | {z } | {z } V1

V2

Vi invariant under a, b, u, whence u|Vi is a (p, q)-sum!! u|V1 exceptional, u|V2 regular. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices



Fp,q = t 2 − σt + (x1 + y1 )(x2 + y2 ) t 2 − σt + (x1 + y2 )(x2 + y1 )

so

Fp,q (u) = Λ(v) for

Λ := t + (x1 + y1 )(x2 + y2 ) t + (x1 + y2 )(x2 + y1 ) ∈ F[t]. For large enough N V = Ker Fp,q (u)N ⊕ Im Fp,q (u)N . | {z } | {z } V1

V2

Vi invariant under a, b, u, whence u|Vi is a (p, q)-sum!! u|V1 exceptional, u|V2 regular. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices



Fp,q = t 2 − σt + (x1 + y1 )(x2 + y2 ) t 2 − σt + (x1 + y2 )(x2 + y1 )

so

Fp,q (u) = Λ(v) for

Λ := t + (x1 + y1 )(x2 + y2 ) t + (x1 + y2 )(x2 + y1 ) ∈ F[t]. For large enough N V = Ker Fp,q (u)N ⊕ Im Fp,q (u)N . | {z } | {z } V1

V2

Vi invariant under a, b, u, whence u|Vi is a (p, q)-sum!! u|V1 exceptional, u|V2 regular. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices



Fp,q = t 2 − σt + (x1 + y1 )(x2 + y2 ) t 2 − σt + (x1 + y2 )(x2 + y1 )

so

Fp,q (u) = Λ(v) for

Λ := t + (x1 + y1 )(x2 + y2 ) t + (x1 + y2 )(x2 + y1 ) ∈ F[t]. For large enough N V = Ker Fp,q (u)N ⊕ Im Fp,q (u)N . | {z } | {z } V1

V2

Vi invariant under a, b, u, whence u|Vi is a (p, q)-sum!! u|V1 exceptional, u|V2 regular. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices



Fp,q = t 2 − σt + (x1 + y1 )(x2 + y2 ) t 2 − σt + (x1 + y2 )(x2 + y1 )

so

Fp,q (u) = Λ(v) for

Λ := t + (x1 + y1 )(x2 + y2 ) t + (x1 + y2 )(x2 + y1 ) ∈ F[t]. For large enough N V = Ker Fp,q (u)N ⊕ Im Fp,q (u)N . | {z } | {z } V1

V2

Vi invariant under a, b, u, whence u|Vi is a (p, q)-sum!! u|V1 exceptional, u|V2 regular. Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Quick reminder on the Frobenius canonical form Companion matrix associated with a monic polynomial r = t k − ak −1 t k −1 − · · · − a0 :   0 (0) a0 1 0 a1      . . . . . . . . .  C(r ) :=   ∈ Mk (F). 0   .. . ..  . 0 ak −2  (0) · · · 0 1 ak −1 Theorem (Frobenius canonical form theorem) Every matrix M is similar to C(r1 ) ⊕ · · · ⊕ C(rk ) for a unique list (r1 , . . . , rk ) of non-constant monic polynomials s.t. ri+1 divides ri for all i. The ri ’s are the invariant factors of M.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Analyzing regular (p, q)-sums Reformulation of earlier results: Theorem Assume p, q split over F. Let M ∈ Mn (F) be (p, q)-regular. Then, M is a (p, q)-sum iff each invariant factor of M equals r (t 2 − σt) for some r ∈ F[t]. Extending the ground field, one easily deduces: Theorem If M ∈ Mn (F) is a regular (p, q)-sum then its invariants factors are polynomials in t 2 − σt. Problem: is the converse still true in the non-split case?

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Analyzing regular (p, q)-sums Reformulation of earlier results: Theorem Assume p, q split over F. Let M ∈ Mn (F) be (p, q)-regular. Then, M is a (p, q)-sum iff each invariant factor of M equals r (t 2 − σt) for some r ∈ F[t]. Extending the ground field, one easily deduces: Theorem If M ∈ Mn (F) is a regular (p, q)-sum then its invariants factors are polynomials in t 2 − σt. Problem: is the converse still true in the non-split case?

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Analyzing regular (p, q)-sums Reformulation of earlier results: Theorem Assume p, q split over F. Let M ∈ Mn (F) be (p, q)-regular. Then, M is a (p, q)-sum iff each invariant factor of M equals r (t 2 − σt) for some r ∈ F[t]. Extending the ground field, one easily deduces: Theorem If M ∈ Mn (F) is a regular (p, q)-sum then its invariants factors are polynomials in t 2 − σt. Problem: is the converse still true in the non-split case?

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Proof of necessity in the split case Assume u = a + b with p(a) = q(b) = 0. Suppose u is regular. Note that Im(a − x1 id) ⊂ Ker(a − x2 id) etc. Moreover Ker(a − xi id) ∩ Ker(b − yj id) = {0} for all i, j (because u is regular). Thus: dim V = 2n for some n; ∀(i, j), dim Ker(a − xi id) = dim Ker(b − yj id) = n; ∀(i, j), Ker(a − x2 id) = Im(a − x1 id), etc. V = Ker(a − x1 id) ⊕ Ker(b − y1 id). Choose (e1 , . . . , en ) basis of Ker(b − y1 id), so
(en+1 , . . . , e2n ) := (a − x1 id)(e1 ), . . . , (a − x1 id)(en )

basis of Im(a − x1 id) = Ker(a − x2 id).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Proof of necessity in the split case Assume u = a + b with p(a) = q(b) = 0. Suppose u is regular. Note that Im(a − x1 id) ⊂ Ker(a − x2 id) etc. Moreover Ker(a − xi id) ∩ Ker(b − yj id) = {0} for all i, j (because u is regular). Thus: dim V = 2n for some n; ∀(i, j), dim Ker(a − xi id) = dim Ker(b − yj id) = n; ∀(i, j), Ker(a − x2 id) = Im(a − x1 id), etc. V = Ker(a − x1 id) ⊕ Ker(b − y1 id). Choose (e1 , . . . , en ) basis of Ker(b − y1 id), so
(en+1 , . . . , e2n ) := (a − x1 id)(e1 ), . . . , (a − x1 id)(en )

basis of Im(a − x1 id) = Ker(a − x2 id).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Proof of necessity in the split case Assume u = a + b with p(a) = q(b) = 0. Suppose u is regular. Note that Im(a − x1 id) ⊂ Ker(a − x2 id) etc. Moreover Ker(a − xi id) ∩ Ker(b − yj id) = {0} for all i, j (because u is regular). Thus: dim V = 2n for some n; ∀(i, j), dim Ker(a − xi id) = dim Ker(b − yj id) = n; ∀(i, j), Ker(a − x2 id) = Im(a − x1 id), etc. V = Ker(a − x1 id) ⊕ Ker(b − y1 id). Choose (e1 , . . . , en ) basis of Ker(b − y1 id), so
(en+1 , . . . , e2n ) := (a − x1 id)(e1 ), . . . , (a − x1 id)(en )

basis of Im(a − x1 id) = Ker(a − x2 id).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Proof of necessity in the split case Assume u = a + b with p(a) = q(b) = 0. Suppose u is regular. Note that Im(a − x1 id) ⊂ Ker(a − x2 id) etc. Moreover Ker(a − xi id) ∩ Ker(b − yj id) = {0} for all i, j (because u is regular). Thus: dim V = 2n for some n; ∀(i, j), dim Ker(a − xi id) = dim Ker(b − yj id) = n; ∀(i, j), Ker(a − x2 id) = Im(a − x1 id), etc. V = Ker(a − x1 id) ⊕ Ker(b − y1 id). Choose (e1 , . . . , en ) basis of Ker(b − y1 id), so
(en+1 , . . . , e2n ) := (a − x1 id)(e1 ), . . . , (a − x1 id)(en )

basis of Im(a − x1 id) = Ker(a − x2 id).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Proof of necessity in the split case Assume u = a + b with p(a) = q(b) = 0. Suppose u is regular. Note that Im(a − x1 id) ⊂ Ker(a − x2 id) etc. Moreover Ker(a − xi id) ∩ Ker(b − yj id) = {0} for all i, j (because u is regular). Thus: dim V = 2n for some n; ∀(i, j), dim Ker(a − xi id) = dim Ker(b − yj id) = n; ∀(i, j), Ker(a − x2 id) = Im(a − x1 id), etc. V = Ker(a − x1 id) ⊕ Ker(b − y1 id). Choose (e1 , . . . , en ) basis of Ker(b − y1 id), so
(en+1 , . . . , e2n ) := (a − x1 id)(e1 ), . . . , (a − x1 id)(en )

basis of Im(a − x1 id) = Ker(a − x2 id).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Proof of necessity in the split case Assume u = a + b with p(a) = q(b) = 0. Suppose u is regular. Note that Im(a − x1 id) ⊂ Ker(a − x2 id) etc. Moreover Ker(a − xi id) ∩ Ker(b − yj id) = {0} for all i, j (because u is regular). Thus: dim V = 2n for some n; ∀(i, j), dim Ker(a − xi id) = dim Ker(b − yj id) = n; ∀(i, j), Ker(a − x2 id) = Im(a − x1 id), etc. V = Ker(a − x1 id) ⊕ Ker(b − y1 id). Choose (e1 , . . . , en ) basis of Ker(b − y1 id), so
(en+1 , . . . , e2n ) := (a − x1 id)(e1 ), . . . , (a − x1 id)(en )

basis of Im(a − x1 id) = Ker(a − x2 id).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Proof of necessity in the split case Assume u = a + b with p(a) = q(b) = 0. Suppose u is regular. Note that Im(a − x1 id) ⊂ Ker(a − x2 id) etc. Moreover Ker(a − xi id) ∩ Ker(b − yj id) = {0} for all i, j (because u is regular). Thus: dim V = 2n for some n; ∀(i, j), dim Ker(a − xi id) = dim Ker(b − yj id) = n; ∀(i, j), Ker(a − x2 id) = Im(a − x1 id), etc. V = Ker(a − x1 id) ⊕ Ker(b − y1 id). Choose (e1 , . . . , en ) basis of Ker(b − y1 id), so
(en+1 , . . . , e2n ) := (a − x1 id)(e1 ), . . . , (a − x1 id)(en )

basis of Im(a − x1 id) = Ker(a − x2 id).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Proof of necessity in the split case Assume u = a + b with p(a) = q(b) = 0. Suppose u is regular. Note that Im(a − x1 id) ⊂ Ker(a − x2 id) etc. Moreover Ker(a − xi id) ∩ Ker(b − yj id) = {0} for all i, j (because u is regular). Thus: dim V = 2n for some n; ∀(i, j), dim Ker(a − xi id) = dim Ker(b − yj id) = n; ∀(i, j), Ker(a − x2 id) = Im(a − x1 id), etc. V = Ker(a − x1 id) ⊕ Ker(b − y1 id). Choose (e1 , . . . , en ) basis of Ker(b − y1 id), so
(en+1 , . . . , e2n ) := (a − x1 id)(e1 ), . . . , (a − x1 id)(en )

basis of Im(a − x1 id) = Ker(a − x2 id).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

In B := (e1 , . . . , e2n ),   x1 In 0n MB (a) = In x2 In



 y1 In N MB (b) = . 0n y2 In

By the translation trick, we can assume x1 = y1 = 0. In that case   0n N =: K (N). MB (a + b) = In σIn Note: if N ≃ N ′ then K (N) ≃ K (N ′ ); if N = N1 ⊕ · · · ⊕ Nk then K (N) ≃ K (N1 ) ⊕ · · · ⊕ K (Nk ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

In B := (e1 , . . . , e2n ),   x1 In 0n MB (a) = In x2 In



 y1 In N MB (b) = . 0n y2 In

By the translation trick, we can assume x1 = y1 = 0. In that case   0n N =: K (N). MB (a + b) = In σIn Note: if N ≃ N ′ then K (N) ≃ K (N ′ ); if N = N1 ⊕ · · · ⊕ Nk then K (N) ≃ K (N1 ) ⊕ · · · ⊕ K (Nk ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

In B := (e1 , . . . , e2n ),   x1 In 0n MB (a) = In x2 In



 y1 In N MB (b) = . 0n y2 In

By the translation trick, we can assume x1 = y1 = 0. In that case   0n N =: K (N). MB (a + b) = In σIn Note: if N ≃ N ′ then K (N) ≃ K (N ′ ); if N = N1 ⊕ · · · ⊕ Nk then K (N) ≃ K (N1 ) ⊕ · · · ⊕ K (Nk ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

In B := (e1 , . . . , e2n ),   x1 In 0n MB (a) = In x2 In



 y1 In N MB (b) = . 0n y2 In

By the translation trick, we can assume x1 = y1 = 0. In that case   0n N =: K (N). MB (a + b) = In σIn Note: if N ≃ N ′ then K (N) ≃ K (N ′ ); if N = N1 ⊕ · · · ⊕ Nk then K (N) ≃ K (N1 ) ⊕ · · · ⊕ K (Nk ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

In B := (e1 , . . . , e2n ),   x1 In 0n MB (a) = In x2 In



 y1 In N MB (b) = . 0n y2 In

By the translation trick, we can assume x1 = y1 = 0. In that case   0n N =: K (N). MB (a + b) = In σIn Note: if N ≃ N ′ then K (N) ≃ K (N ′ ); if N = N1 ⊕ · · · ⊕ Nk then K (N) ≃ K (N1 ) ⊕ · · · ⊕ K (Nk ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Lemma For any monic r ∈ F[t],

K C(r ) ≃ C r (t 2 − σt) .


Sketch of proof: r (t 2 − σt) is the minimal polynomial of K C(r ) .

If N ≃ C(r1 ) ⊕ · · · ⊕ C(rk ) (r1 , . . . , rk invariant factors of N) then

K (N) ≃ C r1 (t 2 − σt) ⊕ · · · ⊕ C rk (t 2 − σt) .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Lemma For any monic r ∈ F[t],

K C(r ) ≃ C r (t 2 − σt) .


Sketch of proof: r (t 2 − σt) is the minimal polynomial of K C(r ) .

If N ≃ C(r1 ) ⊕ · · · ⊕ C(rk ) (r1 , . . . , rk invariant factors of N) then

K (N) ≃ C r1 (t 2 − σt) ⊕ · · · ⊕ C rk (t 2 − σt) .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Lemma For any monic r ∈ F[t],

K C(r ) ≃ C r (t 2 − σt) .


Sketch of proof: r (t 2 − σt) is the minimal polynomial of K C(r ) .

If N ≃ C(r1 ) ⊕ · · · ⊕ C(rk ) (r1 , . . . , rk invariant factors of N) then

K (N) ≃ C r1 (t 2 − σt) ⊕ · · · ⊕ C rk (t 2 − σt) .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The general case Now, we do not assume that p, q are split! Question: Given r ∈ F[t] monic with degree n, is   0n C(r ) M := In σIn a (p, q)-sum? Note that



 C(r ) 0 M − σM = . 0 C(r ) 2

Hence, if M = A + B with p(A) = q(B) = 0, then with R := F[C(r )], A, B are in M2 (R), and with y := C(r ) ∈ R, AB ⋆ + BA⋆ = −(y + (p(0) + q(0))1R )I2 .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The general case Now, we do not assume that p, q are split! Question: Given r ∈ F[t] monic with degree n, is   0n C(r ) M := In σIn a (p, q)-sum? Note that



 C(r ) 0 M − σM = . 0 C(r ) 2

Hence, if M = A + B with p(A) = q(B) = 0, then with R := F[C(r )], A, B are in M2 (R), and with y := C(r ) ∈ R, AB ⋆ + BA⋆ = −(y + (p(0) + q(0))1R )I2 .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The general case Now, we do not assume that p, q are split! Question: Given r ∈ F[t] monic with degree n, is   0n C(r ) M := In σIn a (p, q)-sum? Note that



 C(r ) 0 M − σM = . 0 C(r ) 2

Hence, if M = A + B with p(A) = q(B) = 0, then with R := F[C(r )], A, B are in M2 (R), and with y := C(r ) ∈ R, AB ⋆ + BA⋆ = −(y + (p(0) + q(0))1R )I2 .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The general case Now, we do not assume that p, q are split! Question: Given r ∈ F[t] monic with degree n, is   0n C(r ) M := In σIn a (p, q)-sum? Note that



 C(r ) 0 M − σM = . 0 C(r ) 2

Hence, if M = A + B with p(A) = q(B) = 0, then with R := F[C(r )], A, B are in M2 (R), and with y := C(r ) ∈ R, AB ⋆ + BA⋆ = −(y + (p(0) + q(0))1R )I2 .

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The key algebra

Let R be a commutative F-algebra (with unity). Let x ∈ R. Consider the quotient algebra WR (p, q, x) := R[a, b]/(p(a), q(b), ab ⋆ + ba⋆ − x). Question: Is there a morphism of R-algebras from WR (p, q, x) to M2 (R)? Idea: study the structure of WR (p, q, x). Basic case: R = F[t]/(r n ) where r ∈ F[t] monic and irreducible, n > 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The key algebra

Let R be a commutative F-algebra (with unity). Let x ∈ R. Consider the quotient algebra WR (p, q, x) := R[a, b]/(p(a), q(b), ab ⋆ + ba⋆ − x). Question: Is there a morphism of R-algebras from WR (p, q, x) to M2 (R)? Idea: study the structure of WR (p, q, x). Basic case: R = F[t]/(r n ) where r ∈ F[t] monic and irreducible, n > 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The key algebra

Let R be a commutative F-algebra (with unity). Let x ∈ R. Consider the quotient algebra WR (p, q, x) := R[a, b]/(p(a), q(b), ab ⋆ + ba⋆ − x). Question: Is there a morphism of R-algebras from WR (p, q, x) to M2 (R)? Idea: study the structure of WR (p, q, x). Basic case: R = F[t]/(r n ) where r ∈ F[t] monic and irreducible, n > 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The key algebra

Let R be a commutative F-algebra (with unity). Let x ∈ R. Consider the quotient algebra WR (p, q, x) := R[a, b]/(p(a), q(b), ab ⋆ + ba⋆ − x). Question: Is there a morphism of R-algebras from WR (p, q, x) to M2 (R)? Idea: study the structure of WR (p, q, x). Basic case: R = F[t]/(r n ) where r ∈ F[t] monic and irreducible, n > 0.

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Basic structure of WR (p, q, x) Set A := a, B := b and C := AB. WR (p, q, x) free R-module with basis (1W , A, B, C). Basic computation: C 2 = (tr p tr q − x)C − p(0)q(0)1W . In fact, every element of WR (p, q, x) is quadratic over R! Trace map: Tr : WR (p, q, x) → R morphism of R-modules s.t. Tr(1W ) = 2, Tr(A) = tr(p), Tr(B) = tr(q), Tr(C) = (tr p)(tr q)−x. Conjugation: M ⋆ := Tr(M)1W − M. Norm: ∈R

z }| { ∀M ∈ WR (p, q, x), MM = M M = N(M) .1W ⋆

⋆

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Basic structure of WR (p, q, x) Set A := a, B := b and C := AB. WR (p, q, x) free R-module with basis (1W , A, B, C). Basic computation: C 2 = (tr p tr q − x)C − p(0)q(0)1W . In fact, every element of WR (p, q, x) is quadratic over R! Trace map: Tr : WR (p, q, x) → R morphism of R-modules s.t. Tr(1W ) = 2, Tr(A) = tr(p), Tr(B) = tr(q), Tr(C) = (tr p)(tr q)−x. Conjugation: M ⋆ := Tr(M)1W − M. Norm: ∈R

z }| { ∀M ∈ WR (p, q, x), MM = M M = N(M) .1W ⋆

⋆

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Basic structure of WR (p, q, x) Set A := a, B := b and C := AB. WR (p, q, x) free R-module with basis (1W , A, B, C). Basic computation: C 2 = (tr p tr q − x)C − p(0)q(0)1W . In fact, every element of WR (p, q, x) is quadratic over R! Trace map: Tr : WR (p, q, x) → R morphism of R-modules s.t. Tr(1W ) = 2, Tr(A) = tr(p), Tr(B) = tr(q), Tr(C) = (tr p)(tr q)−x. Conjugation: M ⋆ := Tr(M)1W − M. Norm: ∈R

z }| { ∀M ∈ WR (p, q, x), MM = M M = N(M) .1W ⋆

⋆

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Basic structure of WR (p, q, x) Set A := a, B := b and C := AB. WR (p, q, x) free R-module with basis (1W , A, B, C). Basic computation: C 2 = (tr p tr q − x)C − p(0)q(0)1W . In fact, every element of WR (p, q, x) is quadratic over R! Trace map: Tr : WR (p, q, x) → R morphism of R-modules s.t. Tr(1W ) = 2, Tr(A) = tr(p), Tr(B) = tr(q), Tr(C) = (tr p)(tr q)−x. Conjugation: M ⋆ := Tr(M)1W − M. Norm: ∈R

z }| { ∀M ∈ WR (p, q, x), MM = M M = N(M) .1W ⋆

⋆

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Basic structure of WR (p, q, x) Set A := a, B := b and C := AB. WR (p, q, x) free R-module with basis (1W , A, B, C). Basic computation: C 2 = (tr p tr q − x)C − p(0)q(0)1W . In fact, every element of WR (p, q, x) is quadratic over R! Trace map: Tr : WR (p, q, x) → R morphism of R-modules s.t. Tr(1W ) = 2, Tr(A) = tr(p), Tr(B) = tr(q), Tr(C) = (tr p)(tr q)−x. Conjugation: M ⋆ := Tr(M)1W − M. Norm: ∈R

z }| { ∀M ∈ WR (p, q, x), MM = M M = N(M) .1W ⋆

⋆

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Basic structure of WR (p, q, x) Set A := a, B := b and C := AB. WR (p, q, x) free R-module with basis (1W , A, B, C). Basic computation: C 2 = (tr p tr q − x)C − p(0)q(0)1W . In fact, every element of WR (p, q, x) is quadratic over R! Trace map: Tr : WR (p, q, x) → R morphism of R-modules s.t. Tr(1W ) = 2, Tr(A) = tr(p), Tr(B) = tr(q), Tr(C) = (tr p)(tr q)−x. Conjugation: M ⋆ := Tr(M)1W − M. Norm: ∈R

z }| { ∀M ∈ WR (p, q, x), MM = M M = N(M) .1W ⋆

⋆

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Basic structure of WR (p, q, x) Set A := a, B := b and C := AB. WR (p, q, x) free R-module with basis (1W , A, B, C). Basic computation: C 2 = (tr p tr q − x)C − p(0)q(0)1W . In fact, every element of WR (p, q, x) is quadratic over R! Trace map: Tr : WR (p, q, x) → R morphism of R-modules s.t. Tr(1W ) = 2, Tr(A) = tr(p), Tr(B) = tr(q), Tr(C) = (tr p)(tr q)−x. Conjugation: M ⋆ := Tr(M)1W − M. Norm: ∈R

z }| { ∀M ∈ WR (p, q, x), MM = M M = N(M) .1W ⋆

⋆

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

The Norm (continued) N(α1 + βA + γB + δC) =α α + (tr p)β + (tr q)γ + ((tr p)(tr q) − x)δ
+ β p(0)β + xγ + p(0)(tr q)δ + q(0)β 2
+ (tr p)q(0) γ + p(0)q(0)δ δ. Theorem If R is a field, then N degenerate iff x = x1 y1 + x2 y2 for some x1 , x2 , y1 , y2 in R s.t. p = (t − x1 )(t − x2 ) and q = (t − y1 )(t − y2 ). If R = F[t]/(r ) with r irreducible, and x := −t − p(0) − q(0), then N non-degenerate iff r (t 2 − σt) has no root in Root(p) + Root(q). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices




The Norm (continued) N(α1 + βA + γB + δC) =α α + (tr p)β + (tr q)γ + ((tr p)(tr q) − x)δ
+ β p(0)β + xγ + p(0)(tr q)δ + q(0)β 2
+ (tr p)q(0) γ + p(0)q(0)δ δ. Theorem If R is a field, then N degenerate iff x = x1 y1 + x2 y2 for some x1 , x2 , y1 , y2 in R s.t. p = (t − x1 )(t − x2 ) and q = (t − y1 )(t − y2 ). If R = F[t]/(r ) with r irreducible, and x := −t − p(0) − q(0), then N non-degenerate iff r (t 2 − σt) has no root in Root(p) + Root(q). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices




The Norm (continued) N(α1 + βA + γB + δC) =α α + (tr p)β + (tr q)γ + ((tr p)(tr q) − x)δ
+ β p(0)β + xγ + p(0)(tr q)δ + q(0)β 2
+ (tr p)q(0) γ + p(0)q(0)δ δ. Theorem If R is a field, then N degenerate iff x = x1 y1 + x2 y2 for some x1 , x2 , y1 , y2 in R s.t. p = (t − x1 )(t − x2 ) and q = (t − y1 )(t − y2 ). If R = F[t]/(r ) with r irreducible, and x := −t − p(0) − q(0), then N non-degenerate iff r (t 2 − σt) has no root in Root(p) + Root(q). Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices




Structure theorems Theorem If R is a field and N is non-degenerate then WR (p, q, x) is a quaternion algebra. In particular: If N non-isotropic then WR (p, q, x) skew field. If N isotropic then WR (p, q, x) ≃ M2 (R). Theorem Let r irreducible. Let n > 0 and x ∈ F[t]. Set L := F[t]/(r ) and R := F[t]/(r n ). If the norm of WL (p, q, x) is non-degenerate and isotropic then WR (p, q, x) ≃ M2 (R).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Structure theorems Theorem If R is a field and N is non-degenerate then WR (p, q, x) is a quaternion algebra. In particular: If N non-isotropic then WR (p, q, x) skew field. If N isotropic then WR (p, q, x) ≃ M2 (R). Theorem Let r irreducible. Let n > 0 and x ∈ F[t]. Set L := F[t]/(r ) and R := F[t]/(r n ). If the norm of WL (p, q, x) is non-degenerate and isotropic then WR (p, q, x) ≃ M2 (R).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Structure theorems Theorem If R is a field and N is non-degenerate then WR (p, q, x) is a quaternion algebra. In particular: If N non-isotropic then WR (p, q, x) skew field. If N isotropic then WR (p, q, x) ≃ M2 (R). Theorem Let r irreducible. Let n > 0 and x ∈ F[t]. Set L := F[t]/(r ) and R := F[t]/(r n ). If the norm of WL (p, q, x) is non-degenerate and isotropic then WR (p, q, x) ≃ M2 (R).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Structure theorems Theorem If R is a field and N is non-degenerate then WR (p, q, x) is a quaternion algebra. In particular: If N non-isotropic then WR (p, q, x) skew field. If N isotropic then WR (p, q, x) ≃ M2 (R). Theorem Let r irreducible. Let n > 0 and x ∈ F[t]. Set L := F[t]/(r ) and R := F[t]/(r n ). If the norm of WL (p, q, x) is non-degenerate and isotropic then WR (p, q, x) ≃ M2 (R).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Structure theorems Theorem If R is a field and N is non-degenerate then WR (p, q, x) is a quaternion algebra. In particular: If N non-isotropic then WR (p, q, x) skew field. If N isotropic then WR (p, q, x) ≃ M2 (R). Theorem Let r irreducible. Let n > 0 and x ∈ F[t]. Set L := F[t]/(r ) and R := F[t]/(r n ). If the norm of WL (p, q, x) is non-degenerate and isotropic then WR (p, q, x) ≃ M2 (R).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Conclusion: indecomposable regular (p, q)-sums

Theorem The two types of indecomposable regular (p, q)-sums: C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) isotropic where L := F[t]/(r ). C(r n (t 2 − σt)) ⊕ C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) non-isotropic where L := F[t]/(r ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Conclusion: indecomposable regular (p, q)-sums

Theorem The two types of indecomposable regular (p, q)-sums: C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) isotropic where L := F[t]/(r ). C(r n (t 2 − σt)) ⊕ C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) non-isotropic where L := F[t]/(r ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Conclusion: indecomposable regular (p, q)-sums

Theorem The two types of indecomposable regular (p, q)-sums: C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) isotropic where L := F[t]/(r ). C(r n (t 2 − σt)) ⊕ C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) non-isotropic where L := F[t]/(r ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Conclusion: indecomposable regular (p, q)-sums

Theorem The two types of indecomposable regular (p, q)-sums: C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) isotropic where L := F[t]/(r ). C(r n (t 2 − σt)) ⊕ C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) non-isotropic where L := F[t]/(r ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

Conclusion: indecomposable regular (p, q)-sums

Theorem The two types of indecomposable regular (p, q)-sums: C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) isotropic where L := F[t]/(r ). C(r n (t 2 − σt)) ⊕ C(r n (t 2 − σt)), r ∈ F[t] irreducible, n > 0, r (t 2 − σt) regular, the norm of WL (p, q, −t − p(0) − q(0)) non-isotropic where L := F[t]/(r ).

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

More details in

C. de Seguins Pazzis, The sum and the product of two quadratic matrices, arXiv preprint (2017).

Thank you for your attention!

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices

More details in

C. de Seguins Pazzis, The sum and the product of two quadratic matrices, arXiv preprint (2017).

Thank you for your attention!

Clément de Seguins Pazzis (Lycée Privé Sainte-Geneviève) (Université Sums deand Versailles-Saint-Quentin-en-Yvelines) products of quadratic matrices