A Proof of a Generalized Lax Conjecture for Numerical Ranges - - PowerPoint PPT Presentation

A Proof of a Generalized Lax Conjecture for Numerical Ranges

Kristin A. Camenga (Juniata College) Patrick X. Rault (University of Arizona)

Supported by the American Institute of Mathematics. Joint work with Louis Deaett, Tsvetanka Sendova, Ilya Spitkovsky, and Rebekah Yates.

JMM, January 2018

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Numerical Range

Definition: Numerical Range Let A ∈ Mn(C). Then the numerical range of A is given by W (A) = {⟨Av,v⟩ ∶ v ∈ Cn,∥v∥ = 1} = {v∗Av ∶ v ∈ Cn,v∗v = 1}.

v v↦v*Av Sn ℂ

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Numerical Range

Definition: Numerical Range Let A ∈ Mn(C). Then the numerical range of A is given by W (A) = {⟨Av,v⟩ ∶ v ∈ Cn,∥v∥ = 1} = {v∗Av ∶ v ∈ Cn,v∗v = 1}. Kippenhahn curves Let H1 = A+A∗

2

and H2 = A−A∗

2i

. Then define FA(x ∶ y ∶ t) = det(xH1 + yH2 + tI), ΓFA ∶ FA(x ∶ y ∶ t) = 0, and Γ ˆ

FA its dual.

Then W (A) = Conv(Γ ˆ

FA).

Duality: ax + by + ct = 0 is tangent to ΓFA iff (a ∶ b ∶ c) ∈ Γ ˆ

FA.

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Numerical Range

Kippenhahn curves Let H1 = A+A∗

2

and H2 = A−A∗

2i

. Then define FA(x ∶ y ∶ t) = det(xH1 + yH2 + tI), ΓFA ∶ FA(x ∶ y ∶ t) = 0, and Γ ˆ

FA its dual.

Then W (A) = Conv(Γ ˆ

FA).

Duality: ax + by + ct = 0 is tangent to ΓFA iff (a ∶ b ∶ c) ∈ Γ ˆ

FA.

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• 3
• 2
• 1

1 2

• 3
• 2
• 1

1 2 3

Figure: ΓFA for A ∈ M4(C)

   

• 1

1 2

• 2
• 1

1 2

Figure: Γ ˆ

FA and W (A)

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Numerical Range

Kippenhahn curves Let H1 = A+A∗

2

and H2 = A−A∗

2i

. Then define FA(x ∶ y ∶ t) = det(xH1 + yH2 + tI), ΓFA ∶ FA(x ∶ y ∶ t) = 0, and Γ ˆ

FA its dual.

Then W (A) = Conv(Γ ˆ

FA).

Duality: ax + by + ct = 0 is tangent to ΓFA iff (a ∶ b ∶ c) ∈ Γ ˆ

FA.

FA is hyperbolic of degree n with respect to (0,0,1) Definition: p is hyperbolic with respect to (0,0,1) iff p(0,0,1) ≠ 0 and ∀(a,b,c) ∈ R3 we have that the polynomial p(a,b,c − t) in t has real roots.

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Unitarily Reducible Theorem

Theorem: Kippenhahn (1951) Let A ∈ Mn(C) with A = A1 ⊕ A2. Then: W (A) = Conv(W (A1),W (A2)).

   

1 2 3 4

• ⅈ

ⅈ

Proof Sketch Recall: FA(x ∶ y ∶ t) = det(xH1 + yH2 + tI) A = A1 ⊕ A2 ⇒ FA = FA1 ⋅ FA2. So ΓFA = ΓFA1 ∪ ΓFA2. W (A) = Conv(Γ ˆ

FA) = Conv(Γ ˆ FA1 ∪ Γ ˆ FA2) =

Conv(W (A1) ∪ W (A2)).

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Unitarily Reducible Theorem

Theorem: Kippenhahn (1951) Let A ∈ Mn(C) with A = A1 ⊕ A2. Then: W (A) = Conv(W (A1),W (A2)).

   

1 2 3 4

• ⅈ

ⅈ

Proof Sketch Recall: FA(x ∶ y ∶ t) = det(xH1 + yH2 + tI) A = A1 ⊕ A2 ⇒ FA = FA1 ⋅ FA2. ... Questions: What if FA is reducible and A is not? Do the factors of FA correspond to some numerical ranges of (smaller) matrices?

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Which polynomials yield matrices?

Let Sn(X) denote the set of symmetric matrices in Mn(X). Lax Conjecture (1958) & Lewis, Parrilo, Ramana Theorem (2005) The following are equivalent: p ∈ R[x,y,z], hyperbolic of degree n with respect to (0,0,1) and p(0,0,1) = 1. ∃C,D ∈ Sn(R) such that p(x,y,z) = det(xC + yD + zIn). Question: is Conv(Γˆ

p) the numerical range of some matrix?

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Which polynomials yield matrices?

Let Sn(X) denote the set of symmetric matrices in Mn(X). Lax Conjecture (1958) & Lewis, Parrilo, Ramana Theorem (2005) The following are equivalent: p ∈ R[x,y,z], hyperbolic of degree n with respect to (0,0,1) and p(0,0,1) = 1. ∃C,D ∈ Sn(R) such that p(x,y,z) = det(xC + yD + zIn). Question: is Conv(Γˆ

p) the numerical range of some matrix?

Helton, Spitkovsky Theorem (2012) Let A ∈ Mn(C). Then there exists B ∈ Sn(C) such that W (A) = W (B). CDRSSY Theorem Let A ∈ Mn(C) and let G ∈ R[x,y,t] be a factor of FA (possibly G = FA). Then ∃B ∈ Sn(C) such that FB = G. And W (B) = Conv(Γ ˆ

G).

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Which polynomials yield matrices?

Let Sn(X) denote the set of symmetric matrices in Mn(X). CDRSSY Theorem Let A ∈ Mn(C) and let G ∈ R[x,y,t] be a factor of FA (possibly G = FA). Then ∃B ∈ Sn(C) such that FB = G. And W (B) = Conv(Γ ˆ

G).

Proof Since roots of G are roots of FA, G is also hyperbolic w.r.t. (0,0,1). Replace G by

G G(0,0,1) and let d = deg G. LPR2005 gives C,D ∈ Sd(R) for

which G(x,y,t) = det(xC + yD + tId). Let B = C + Di, and note that C = B+B∗

2

and D = B−B∗

2i

. Since C,D ∈ Sd(R), we have B ∈ Sd(C).

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Which polynomials yield matrices?

CDRSSY Theorem Let A ∈ Mn(C) and let G ∈ R[x,y,t] be a factor of FA (possibly G = FA). Then ∃B ∈ Sn(C) such that FB = G. And W (B) = Conv(Γ ˆ

G).

Example: Let A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

4 1 4 1 1 4 1 4 1 4 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .

• 1

1

• ⅈ

ⅈ

Figure: Γ ˆ

FA &

W (A) = W (A1 ⊕ A2 ⊕ A3 ⊕ A4)

FA(x,y,t) = − t

32(25x2 + 9y 2 − 2t2)

(25x2(2 + √ 2) + 9y 2(2 + √ 2) − 4t2)(25x2(2 − √ 2) + 9y 2(2 − √ 2) − 4t2).

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Questions?

Kristin Camenga camenga@juniata.edu Patrick Rault rault@email.arizona.edu

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