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A 2D-DFT based method to compute the Bezoutian and a link to Lyapunov equations Chayan Bhawal, Debasattam Pal and Madhu N. Belur Control and Computing Department of Electrical Engineering IIT Bombay Indian Control Conference, Guwahati January

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A 2D-DFT based method to compute the Bezoutian and a link to Lyapunov equations

Chayan Bhawal, Debasattam Pal and Madhu N. Belur Control and Computing Department of Electrical Engineering IIT Bombay Indian Control Conference, Guwahati January 6, 2017

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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Introduction

Introduction

p, q are polynomials: p(x)q(y) + p(y)q(x) x + y =: ˜ B(x, y) p(x)q(y) − p(y)q(x) x − y =: b(x, y) Bezoutian Stability analysis Riccati equation solutions Storage functions Polynomial coprimeness

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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Introduction

Introduction

p, q are polynomials: p(x)q(y) + p(y)q(x) x + y =: ˜ B(x, y) p(x)q(y) − p(y)q(x) x − y =: b(x, y) Bezoutian Stability analysis Riccati equation solutions Storage functions Polynomial coprimeness

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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Introduction

Introduction

p, q are polynomials: p(x)q(y) + p(y)q(x) x + y =: ˜ B(x, y) p(x)q(y) − p(y)q(x) x − y =: b(x, y) Stability analysis Riccati equation solutions Storage functions Polynomial coprimeness Bezoutian

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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Introduction

Bezoutian b(x, y)

p, q are polynomials: p(x)q(y) + p(y)q(x) x + y =: b(x, y) p(x)q(y) − p(y)q(x) x − y =: b(x, y) Stability analysis Riccati equation solutions Storage functions Polynomial coprimeness Bezoutian

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 6

Introduction

Bezoutian b(x, y)

p, q are polynomials: φ(x, y) x + y =: b(x, y) p(x)q(y) − p(y)q(x) x − y =: b(x, y) Stability analysis Riccati equation solutions Storage functions Polynomial coprimeness Bezoutian

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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Introduction

Bezoutian b(x, y)

p, q are polynomials: φ(x, y) = (x + y)b(x, y) p(x)q(y) − p(y)q(x) x − y =: b(x, y) Stability analysis Riccati equation solutions Storage functions Polynomial coprimeness Bezoutian

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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Outline

Objective and Outline

p, q are polynomials: φ(x, y) = (x + y)b(x, y) OBJECTIVE: To compute the Bezoutian b(x, y). OUTLINE

1 2D-DFT based method to compute Bezoutian. 2 Bezoutian and link to Lyapunov equation. 3 Lyapunov equation and its link to two variable polynomials. Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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Outline

Objective and Outline

p, q are polynomials: φ(x, y) = (x + y)b(x, y) OBJECTIVE: To compute the Bezoutian b(x, y). OUTLINE

1 2D-DFT based method to compute Bezoutian. 2 Bezoutian and link to Lyapunov equation. 3 Lyapunov equation and its link to two variable polynomials. Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

The Algorithm

Let X : =        1 x x2 . . . xN−1        , Y : =        1 y y2 . . . yN−1        Bezoutian: (x + y)b(x, y) = φ(x, y) φ(x, y)= XT ΦY, Φ ∈ RN×N (x + y) = XT J

  • Y =: XT RY, J =

1 1

  • ∈ R2×2

b(x, y) =XT ˜

B

  • Y=: XT BY, ˜

B ∈ R(N−1)×(N−1)

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

The Algorithm

Let X : =        1 x x2 . . . xN−1        , Y : =        1 y y2 . . . yN−1        Bezoutian: (x + y)b(x, y) = φ(x, y) φ(x, y) = XT ΦY, Φ ∈ RN×N (x + y) = XT J

  • Y =: XT RY, J =

1 1

  • ∈ R2×2

b(x, y) =XT ˜

B

  • Y=: XT BY, ˜

B ∈ R(N−1)×(N−1)

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

The Algorithm

Let X : =        1 x x2 . . . xN−1        , Y : =        1 y y2 . . . yN−1        Bezoutian: (x + y)b(x, y) = φ(x, y) φ(x, y) = XT ΦY, Φ ∈ RN×N (x + y) = XT J

  • Y =: XT RY, J =

1 1

  • ∈ R2×2

b(x, y) =XT ˜

B

  • Y=: XT BY, ˜

B ∈ R(N−1)×(N−1)

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

The Algorithm

Let X : =        1 x x2 . . . xN−1        , Y : =        1 y y2 . . . yN−1        Bezoutian: (x + y)b(x, y) = φ(x, y) φ(x, y) = XT ΦY, Φ ∈ RN×N (x + y) = XT J

  • Y =: XT RY, J =

1 1

  • ∈ R2×2

b(x, y) =XT ˜

B

  • Y=: XT BY, ˜

B ∈ R(N−1)×(N−1)

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

The Algorithm

(x + y)b(x, y) = φ(x, y) = ⇒

  • XT RY

XT BY

  • =
  • XT ΦY
  • # Objective: Find B.

# Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F(R) ⊗ F(B) = F(Φ) where ⊗ means elementwise multiplication.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

The Algorithm

(x + y)b(x, y) = φ(x, y) = ⇒

  • XT RY

XT BY

  • =
  • XT ΦY
  • # Objective: Find B.

# Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F(R) ⊗ F(B) = F(Φ) where ⊗ means elementwise multiplication.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

The Algorithm

(x + y)b(x, y) = φ(x, y) = ⇒

  • XT RY

XT BY

  • =
  • XT ΦY
  • # Objective: Find B.

# Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F(R) ⊗ F(B) = F(Φ) where ⊗ means elementwise multiplication.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 17

2D-DFT and Bezoutian

The Algorithm

(x + y)b(x, y) = φ(x, y) = ⇒

  • XT RY

XT BY

  • =
  • XT ΦY
  • # Objective: Find B.

# Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F(R) ⊗ F(B) = F(Φ) where ⊗ means elementwise multiplication.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

The Algorithm

(x + y)b(x, y) = φ(x, y) = ⇒

  • XT RY

XT BY

  • =
  • XT ΦY
  • # Objective: Find B.

# Different problems have different coefficient matrix Φ. # R remains fixed. Two variable polynomial multiplication ⇔ 2D-Convolution. R ⋆ B = Φ where ⋆ means 2D-convolution. 2D-convolution ⇔ Elementwise multiplication F(R) ⊗ F(B) = F(Φ) where ⊗ means elementwise multiplication.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

B = F −1 F(Φ)./F(R)

  • .

The algorithm fails when any element of F(R) is zero. (k, ℓ)th element of F(R) is e−j 2π

N k + e−j 2π N ℓ i.e. pairwise sum of

roots of unity.

1

  • 1

N is even

1

N is odd N even case: Zero padding required in R and Φ to make them odd.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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2D-DFT and Bezoutian

B = F −1 F(Φ)./F(R)

  • .

The algorithm fails when any element of F(R) is zero. (k, ℓ)th element of F(R) is e−j 2π

N k + e−j 2π N ℓ i.e. pairwise sum of

roots of unity.

1

  • 1

N is even

1

N is odd N even case: Zero padding required in R and Φ to make them odd.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 21

2D-DFT and Bezoutian

B = F −1 F(Φ)./F(R)

  • .

The algorithm fails when any element of F(R) is zero. (k, ℓ)th element of F(R) is e−j 2π

N k + e−j 2π N ℓ i.e. pairwise sum of

roots of unity.

1

  • 1

N is even

1

N is odd N even case: Zero padding required in R and Φ to make them odd.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 22

2D-DFT and Bezoutian

B = F −1 F(Φ)./F(R)

  • .

The algorithm fails when any element of F(R) is zero. (k, ℓ)th element of F(R) is e−j 2π

N k + e−j 2π N ℓ i.e. pairwise sum of

roots of unity.

1

  • 1

N is even

1

N is odd N even case: Zero padding required in R and Φ to make them odd.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 23

Bezoutian and Lyapunov operators

Bezoutian and Lyapunov operators

b(x, y) = XT BY and φ(x, y) = XT ΦY (x + y)b(x, y) = φ(x, y) E :=        · · · 1 1 · · · 1 · · · . . . . . . ... . . . . . . · · · 1        ∈ RN×N (call unit cyclic matrix). Then, EB + BET = Φ xb(x, y) + yb(x, y) = XT EBY + XT BET Y.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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Bezoutian and Lyapunov operators

Bezoutian and Lyapunov operators

b(x, y) = XT BY and φ(x, y) = XT ΦY (x + y)b(x, y) = φ(x, y) E :=        · · · 1 1 · · · 1 · · · . . . . . . ... . . . . . . · · · 1        ∈ RN×N (call unit cyclic matrix). Then, EB + BET = Φ xb(x, y) + yb(x, y) = XT EBY + XT BET Y.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 25

Bezoutian and Lyapunov operators

Bezoutian and Lyapunov operators

b(x, y) = XT BY and φ(x, y) = XT ΦY (x + y)b(x, y) = φ(x, y) E :=        · · · 1 1 · · · 1 · · · . . . . . . ... . . . . . . · · · 1        ∈ RN×N (call unit cyclic matrix). Then, EB + BET = Φ xb(x, y) + yb(x, y) = XT EBY + XT BET Y.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 26

Bezoutian and Lyapunov operators

Bezoutian and Lyapunov operators

b(x, y) = XT BY and φ(x, y) = XT ΦY (x + y)b(x, y) = φ(x, y) E :=        · · · 1 1 · · · 1 · · · . . . . . . ... . . . . . . · · · 1        ∈ RN×N (call unit cyclic matrix). Then, EB + BET = Φ xb(x, y) + yb(x, y) = XT EBY + XT BET Y.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 27

Bezoutian and Lyapunov operators

Bezoutian and Lyapunov operators

b(x, y) = XT BY and φ(x, y) = XT ΦY (x + y)b(x, y) = φ(x, y) E :=        · · · 1 1 · · · 1 · · · . . . . . . ... . . . . . . · · · 1        (call unit cyclic matrix). Then, EB + BET = Φ Relation between 2D-DFT and Lyapunov equation?

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 28

Bezoutian and Lyapunov operators

Lyapunov operator

LA(P) = AP + PAT where A ∈ RN×N, P ∈ CN×N. Eigenvalues of A be {λ0, λ1, . . . , λN−1} corresponding to eigenvectors {v0, v1, . . . , vN−1}. LA(viv∗

j ) = (λi + λ∗ j)viv∗ j (compare with Ax = λx).

Eigenmatrix of LA(•) is viv∗

j with respect to eigenvalue (λi + λ∗ j).

For us, A = E i.e. LE(P) = EP + PET . Eigenvalues and eigenvectors of E matters.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 29

Bezoutian and Lyapunov operators

Lyapunov operator

LA(P) = AP + PAT where A ∈ RN×N, P ∈ CN×N. Eigenvalues of A be {λ0, λ1, . . . , λN−1} corresponding to eigenvectors {v0, v1, . . . , vN−1}. LA(viv∗

j ) = (λi + λ∗ j)viv∗ j (compare with Ax = λx).

Eigenmatrix of LA(•) is viv∗

j with respect to eigenvalue (λi + λ∗ j).

For us, A = E i.e. LE(P) = EP + PET . Eigenvalues and eigenvectors of E matters.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 30

Bezoutian and Lyapunov operators

Lyapunov operator

LA(P) = AP + PAT where A ∈ RN×N, P ∈ CN×N. Eigenvalues of A be {λ0, λ1, . . . , λN−1} corresponding to eigenvectors {v0, v1, . . . , vN−1}. LA(viv∗

j ) = (λi + λ∗ j)viv∗ j (compare with Ax = λx).

Eigenmatrix of LA(•) is viv∗

j with respect to eigenvalue (λi + λ∗ j).

For us, A = E i.e. LE(P) = EP + PET . Eigenvalues and eigenvectors of E matters.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 31

Bezoutian and Lyapunov operators

Lyapunov operator

LA(P) = AP + PAT where A ∈ RN×N, P ∈ CN×N. Eigenvalues of A be {λ0, λ1, . . . , λN−1} corresponding to eigenvectors {v0, v1, . . . , vN−1}. LA(viv∗

j ) = (λi + λ∗ j)viv∗ j (compare with Ax = λx).

Eigenmatrix of LA(•) is viv∗

j with respect to eigenvalue (λi + λ∗ j).

For us, A = E i.e. LE(P) = EP + PET . Eigenvalues and eigenvectors of E matters.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 32

Bezoutian and Lyapunov operators

Lyapunov operator

LA(P) = AP + PAT where A ∈ RN×N, P ∈ CN×N. Eigenvalues of A be {λ0, λ1, . . . , λN−1} corresponding to eigenvectors {v0, v1, . . . , vN−1}. LA(viv∗

j ) = (λi + λ∗ j)viv∗ j (compare with Ax = λx).

Eigenmatrix of LA(•) is viv∗

j with respect to eigenvalue (λi + λ∗ j).

For us, A = E i.e. LE(P) = EP + PET . Eigenvalues and eigenvectors of E matters.

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 33

Bezoutian and Lyapunov operators

Bezoutian and Lyapunov operators

(x + y) = XT RY where R = J

  • ∈ RN×N and J =

1 1

  • .

Elements of F(R) are pairwise sum of roots of unity. LE(P) := EP + PET where E ∈ CN×N is the unit cyclic matrix. Eigenvalues of LE(•) are also pairwise sum of roots of unity. Then, Elements of F(R) ≡ corresponding eigenvalues of LE(•). Is there a link between Lyapunov operators and two variable polynomial multiplication in general?

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 34

Bezoutian and Lyapunov operators

Bezoutian and Lyapunov operators

(x + y) = XT RY where R = J

  • ∈ RN×N and J =

1 1

  • .

Elements of F(R) are pairwise sum of roots of unity. LE(P) := EP + PET where E ∈ CN×N is the unit cyclic matrix. Eigenvalues of LE(•) are also pairwise sum of roots of unity. Then, Elements of F(R) ≡ corresponding eigenvalues of LE(•). Is there a link between Lyapunov operators and two variable polynomial multiplication in general?

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

slide-35
SLIDE 35

Bezoutian and Lyapunov operators

Bezoutian and Lyapunov operators

(x + y) = XT RY where R = J

  • ∈ RN×N and J =

1 1

  • .

Elements of F(R) are pairwise sum of roots of unity. LE(P) := EP + PET where E ∈ CN×N is the unit cyclic matrix. Eigenvalues of LE(•) are also pairwise sum of roots of unity. Then, Elements of F(R) ≡ corresponding eigenvalues of LE(•). Is there a link between Lyapunov operators and two variable polynomial multiplication in general?

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

slide-36
SLIDE 36

Bezoutian and Lyapunov operators

Bezoutian and Lyapunov operators

(x + y) = XT RY where R = J

  • ∈ RN×N and J =

1 1

  • .

Elements of F(R) are pairwise sum of roots of unity. LE(P) := EP + PET where E ∈ CN×N is the unit cyclic matrix. Eigenvalues of LE(•) are also pairwise sum of roots of unity. Then, Elements of F(R) ≡ corresponding eigenvalues of LE(•). Is there a link between Lyapunov operators and two variable polynomial multiplication in general?

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 37

Two variable polynomials and Lyapunov operator Main result

Two variable polynomials and Lyapunov operator

LE(P) := EP + PET where E is unit cyclic matrix. Let V ∈ CN×N and v(x, y) := XT V Y. Canonical surjection map Π : C[x, y] − → C[x, y]/A where A := xN − 1, yN − 1. Then, LE(V ) = µV ⇐ ⇒ Π

  • (x + y)v(x, y)
  • = µv(x, y).

A is the ideal generated by (xN − 1), (yN − 1) i.e. the set of all polynomials of the form e(x, y)(xN − 1) + f(x, y)(yN − 1).

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 38

Two variable polynomials and Lyapunov operator Main result

Two variable polynomials and Lyapunov operator

LE(P) := EP + PET where E is unit cyclic matrix. Let V ∈ CN×N and v(x, y) := XT V Y. Canonical surjection map Π : C[x, y] − → C[x, y]/A where A := xN − 1, yN − 1. Then, LE(V ) = µV ⇐ ⇒ Π

  • (x + y)v(x, y)
  • = µv(x, y).

A is the ideal generated by (xN − 1), (yN − 1) i.e. the set of all polynomials of the form e(x, y)(xN − 1) + f(x, y)(yN − 1).

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 39

Two variable polynomials and Lyapunov operator Main result

Two variable polynomials and Lyapunov operator

LE(P) := EP + PET where E is unit cyclic matrix. Let V ∈ CN×N and v(x, y) := XT V Y. Canonical surjection map Π : C[x, y] − → C[x, y]/A where A := xN − 1, yN − 1. Then, LE(V ) = µV ⇐ ⇒ Π

  • (x + y)v(x, y)
  • = µv(x, y).

A is the ideal generated by (xN − 1), (yN − 1) i.e. the set of all polynomials of the form e(x, y)(xN − 1) + f(x, y)(yN − 1).

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 40

Two variable polynomials and Lyapunov operator Main result

Two variable polynomials and Lyapunov operator

LE(P) := EP + PET where E is unit cyclic matrix. Let V ∈ CN×N and v(x, y) := XT V Y. Canonical surjection map Π : C[x, y] − → C[x, y]/A where A := xN − 1, yN − 1. Then, LE(V ) = µV ⇐ ⇒ Π

  • (x + y)v(x, y)
  • = µv(x, y).

A is the ideal generated by (xN − 1), (yN − 1) i.e. the set of all polynomials of the form e(x, y)(xN − 1) + f(x, y)(yN − 1).

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 41

Two variable polynomials and Lyapunov operator Example

Example

Consider E = 1 1

  • ∈ C2×2 .

Consider V = 1 1 1 1

  • . Then LE(V ) = 2V .

v(x, y) = XT V Y = 1 + x + y + xy. (x + y)v(x, y)/x2 − 1, y2 − 1 = (x2 + x2y + xy2 + 2xy + x + y2 + y)/x2 − 1, y2 − 1 = 2 + 2x + 2y + 2xy = 2v(x, y)

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 42

Two variable polynomials and Lyapunov operator Example

Example

Consider E = 1 1

  • ∈ C2×2 .

Consider V = 1 1 1 1

  • . Then LE(V ) = 2V .

v(x, y) = XT V Y = 1 + x + y + xy. (x + y)v(x, y)/x2 − 1, y2 − 1 = (x2 + x2y + xy2 + 2xy + x + y2 + y)/x2 − 1, y2 − 1 = 2 + 2x + 2y + 2xy = 2v(x, y)

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 43

Two variable polynomials and Lyapunov operator Example

Example

Consider E = 1 1

  • ∈ C2×2 .

Consider V = 1 1 1 1

  • . Then LE(V ) = 2V .

v(x, y) = XT V Y = 1 + x + y + xy. (x + y)v(x, y)/x2 − 1, y2 − 1 = (x2 + x2y + xy2 + 2xy + x + y2 + y)/x2 − 1, y2 − 1 = 2 + 2x + 2y + 2xy = 2v(x, y)

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

slide-44
SLIDE 44

Two variable polynomials and Lyapunov operator Example

Example

Consider E = 1 1

  • ∈ C2×2 .

Consider V = 1 1 1 1

  • . Then LE(V ) = 2V .

v(x, y) = XT V Y = 1 + x + y + xy. (x + y)v(x, y)/x2 − 1, y2 − 1 = (x2 + x2y + xy2 + 2xy + x + y2 + y)/x2 − 1, y2 − 1 = 2 + 2x + 2y + 2xy = 2v(x, y)

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 45

Two variable polynomials and Lyapunov operator Example

Two variable polynomials and Lyapunov operator

LE(V ) = µV ⇐ ⇒ Π

  • (x + y)v(x, y)
  • = µv(x, y).

Analogous result: ℓE(p) := Ep where E ∈ CN×N is the unit cyclic matrix Let p ∈ CN×1 and v(x) = XT p π : C[x] − → R[x]/a where a := xN − 1. ℓE(q) = λq ⇐ ⇒ π

  • xv(x)
  • = λ(N−1)v(x).

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 46

Conclusion

Conclusion

1 Reported a method to compute Bezoutian using 2D-DFT. 2 Link between Bezoutian and Lyapunov operator. 3 Two variable interpretation of Lyapunov operator.

THANK YOU Queries?

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 47

Conclusion

2D-Convolution: example

(1 + x + y)(2 + xy) = 1 x T 1 1 1 1 y 1 x T 2 1 1 y

  • = 2 + 2x + 2y + xy + x2y + xy2

We compute: 2D-convolution of coefficient matrices, 1 1 1

  • ⋆

2 1

  • 2D-convolution formula

y[m, n] =

N

  • j=0

N

  • i=0

x[i, j]h[m − i, n − j]

V.Y. Pan, Structured Matrices and Polynomials: Unified Superfast Algorithms, Birkh¨ auser, 2001. Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 48

Conclusion

2D-Convolution: example

(1 + x + y)(2 + xy) = 1 x T 1 1 1 1 y 1 x T 2 1 1 y

  • = 2 + 2x + 2y + xy + x2y + xy2

We compute: 2D-convolution of coefficient matrices, 1 1 1

  • ⋆

2 1

  • 2D-convolution formula

y[m, n] =

N

  • j=0

N

  • i=0

x[i, j]h[m − i, n − j]

V.Y. Pan, Structured Matrices and Polynomials: Unified Superfast Algorithms, Birkh¨ auser, 2001. Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 49

Conclusion

2D-Convolution

Example: (1 + x + y)(2 + xy) = 2 + 2x + 2y + xy + x2y + xy2 1 1 1

  • ⋆

2 1

  • 1

2 1 1 1   2 0 

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 50

Conclusion

2D-Convolution

Example: (1 + x + y)(2 + xy) = 2 + 2x + 2y + xy + x2y + xy2 1 1 1

  • ⋆

2 1

  • 1

2 1 1 1   2 2 0 

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 51

Conclusion

2D-Convolution

Example: (1 + x + y)(2 + xy) = 2 + 2x + 2y + xy + x2y + xy2 1 1 1

  • ⋆

2 1

  • 1

2 1 1 1   2 2 0 

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 52

Conclusion

2D-Convolution

Example: (1 + x + y)(2 + xy) = 2 + 2x + 2y + xy + x2y + xy2 1 1 1

  • ⋆

2 1

  • 1

2 1 1 1   2 2 2  

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 53

Conclusion

2D-Convolution

Example: (1 + x + y)(2 + xy) = 2 + 2x + 2y + xy + x2y + xy2 1 1 1

  • ⋆

2 1

  • 1

2 1 1 1   2 2 2 1  

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 54

Conclusion

2D-Convolution

Example: (1 + x + y)(2 + xy) = 2 + 2x + 2y + xy + x2y + xy2 1 1 1

  • ⋆

2 1

  • 1

2 1 1 1   2 2 2 1 1  

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 55

Conclusion

2D-Convolution

Example: (1 + x + y)(2 + xy) = 2 + 2x + 2y + xy + x2y + xy2 1 1 1

  • ⋆

2 1

  • 1

2 1 1 1   2 2 2 1 1  

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 56

Conclusion

2D-Convolution

Example: (1 + x + y)(2 + xy) = 2 + 2x + 2y + xy + x2y + xy2 1 1 1

  • ⋆

2 1

  • 1

2 1 1 1   2 2 2 1 1 1  

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12

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SLIDE 57

Conclusion

2D-Convolution

Example: (1 + x + y)(2 + xy) = 2 + 2x + 2y + xy + x2y + xy2 1 1 1

  • ⋆

2 1

  • 1

2 1 1 1   2 2 2 1 1 1  

Bhawal, Pal, Belur (IIT Bombay) 2D-DFT, Bezoutian, Lyapunov eqn. Indian Control Conference, GuwahatiJan / 12