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 Camenga & Rault (JC & UA ) Generalized Lax Conjecture 1 / 5

Numerical Range Definition: Numerical Range Let A ∈ M n ( C ) . Then the numerical range of A is given by W ( A ) = {⟨ Av , v ⟩ ∶ v ∈ C n , ∥ v ∥ = 1 } = { v ∗ Av ∶ v ∈ C n , v ∗ v = 1 } . S n v ↦ v * Av ℂ v Camenga & Rault (JC & UA ) Generalized Lax Conjecture 2 / 5

Numerical Range Definition: Numerical Range Let A ∈ M n ( C ) . Then the numerical range of A is given by W ( A ) = {⟨ Av , v ⟩ ∶ v ∈ C n , ∥ v ∥ = 1 } = { v ∗ Av ∶ v ∈ C n , v ∗ v = 1 } . Kippenhahn curves Let H 1 = A + A ∗ and H 2 = A − A ∗ . Then define 2 2 i F A ( x ∶ y ∶ t ) = det ( xH 1 + yH 2 + t I ) , Γ F A ∶ F A ( x ∶ y ∶ t ) = 0, and Γ ˆ F A its dual. Then W ( A ) = Conv ( Γ ˆ F A ) . Duality: ax + by + ct = 0 is tangent to Γ F A iff ( a ∶ b ∶ c ) ∈ Γ ˆ F A . Camenga & Rault (JC & UA ) Generalized Lax Conjecture 2 / 5

Numerical Range Kippenhahn curves Let H 1 = A + A ∗ and H 2 = A − A ∗ . Then define 2 2 i F A ( x ∶ y ∶ t ) = det ( xH 1 + yH 2 + t I ) , Γ F A ∶ F A ( x ∶ y ∶ t ) = 0, and Γ ˆ F A its dual. Then W ( A ) = Conv ( Γ ˆ F A ) . Duality: ax + by + ct = 0 is tangent to Γ F A iff ( a ∶ b ∶ c ) ∈ Γ ˆ F A . 3 2 2  1 1 0 0   - 1 - 1  - 2 - 2 - 3 - 1 0 1 2 - 5 - 4 - 3 - 2 - 1 0 1 2 F A and W ( A ) Figure: Γ ˆ Figure: Γ F A for A ∈ M 4 ( C ) Camenga & Rault (JC & UA ) Generalized Lax Conjecture 2 / 5

Numerical Range Kippenhahn curves Let H 1 = A + A ∗ and H 2 = A − A ∗ . Then define 2 2 i F A ( x ∶ y ∶ t ) = det ( xH 1 + yH 2 + t I ) , Γ F A ∶ F A ( x ∶ y ∶ t ) = 0, and Γ ˆ F A its dual. Then W ( A ) = Conv ( Γ ˆ F A ) . Duality: ax + by + ct = 0 is tangent to Γ F A iff ( a ∶ b ∶ c ) ∈ Γ ˆ F A . F A 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 ) ∈ R 3 we have that the polynomial p ( a , b , c − t ) in t has real roots. Camenga & Rault (JC & UA ) Generalized Lax Conjecture 2 / 5

Unitarily Reducible Theorem Theorem: Kippenhahn (1951) Let A ∈ M n ( C ) with A = A 1 ⊕ A 2 . Then: W ( A ) = Conv ( W ( A 1 ) , W ( A 2 )) . Proof Sketch Recall: F A ( x ∶ y ∶ t ) = det ( xH 1 + yH 2 + t I ) ⅈ  A = A 1 ⊕ A 2 ⇒ F A = F A 1 ⋅ F A 2 . 0   So Γ F A = Γ F A 1 ∪ Γ F A 2 .  W ( A ) = Conv ( Γ ˆ F A ) = Conv ( Γ ˆ F A 1 ∪ Γ ˆ F A 2 ) = - ⅈ Conv ( W ( A 1 ) ∪ W ( A 2 )) . 1 2 3 4 Camenga & Rault (JC & UA ) Generalized Lax Conjecture 3 / 5

Unitarily Reducible Theorem Theorem: Kippenhahn (1951) Let A ∈ M n ( C ) with A = A 1 ⊕ A 2 . Then: W ( A ) = Conv ( W ( A 1 ) , W ( A 2 )) . Proof Sketch Recall: F A ( x ∶ y ∶ t ) = det ( xH 1 + yH 2 + t I ) ⅈ A = A 1 ⊕ A 2 ⇒ F A = F A 1 ⋅ F A 2 .  0   ...  Questions: What if F A is reducible and A is - ⅈ not? Do the factors of F A correspond to some numerical ranges of (smaller) 1 2 3 4 matrices? Camenga & Rault (JC & UA ) Generalized Lax Conjecture 3 / 5

Which polynomials yield matrices? Let S n ( X ) denote the set of symmetric matrices in M n ( 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 ∈ S n ( R ) such that p ( x , y , z ) = det ( xC + yD + z I n ) . Question: is Conv ( Γ ˆ p ) the numerical range of some matrix? Camenga & Rault (JC & UA ) Generalized Lax Conjecture 4 / 5

Which polynomials yield matrices? Let S n ( X ) denote the set of symmetric matrices in M n ( 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 ∈ S n ( R ) such that p ( x , y , z ) = det ( xC + yD + z I n ) . Question: is Conv ( Γ ˆ p ) the numerical range of some matrix? Helton, Spitkovsky Theorem (2012) Let A ∈ M n ( C ) . Then there exists B ∈ S n ( C ) such that W ( A ) = W ( B ) . CDRSSY Theorem Let A ∈ M n ( C ) and let G ∈ R [ x , y , t ] be a factor of F A (possibly G = F A ). Then ∃ B ∈ S n ( C ) such that F B = G . And W ( B ) = Conv ( Γ ˆ G ) . Camenga & Rault (JC & UA ) Generalized Lax Conjecture 4 / 5

Which polynomials yield matrices? Let S n ( X ) denote the set of symmetric matrices in M n ( X ) . CDRSSY Theorem Let A ∈ M n ( C ) and let G ∈ R [ x , y , t ] be a factor of F A (possibly G = F A ). Then ∃ B ∈ S n ( C ) such that F B = G . And W ( B ) = Conv ( Γ ˆ G ) . Proof Since roots of G are roots of F A , G is also hyperbolic w.r.t. ( 0 , 0 , 1 ) . G G ( 0 , 0 , 1 ) and let d = deg G . LPR2005 gives C , D ∈ S d ( R ) for Replace G by which G ( x , y , t ) = det ( xC + yD + t I d ) . Let B = C + Di , and note that C = B + B ∗ and D = B − B ∗ . Since 2 2 i C , D ∈ S d ( R ) , we have B ∈ S d ( C ) . Camenga & Rault (JC & UA ) Generalized Lax Conjecture 4 / 5

Which polynomials yield matrices? CDRSSY Theorem Let A ∈ M n ( C ) and let G ∈ R [ x , y , t ] be a factor of F A (possibly G = F A ). Then ∃ B ∈ S n ( C ) such that F B = G . And W ( B ) = Conv ( Γ ˆ G ) . ⅈ Example: ⎡ ⎤ 0 4 0 0 0 0 0 ⎢ ⎥ 0 ⎢ ⎥ 1 0 4 0 0 0 0 ⎢ ⎥ ⎢ ⎥ 0 1 0 0 0 0 0 ⎢ ⎥ ⎢ ⎥ Let A = . 0 0 1 0 4 0 0 ⎢ ⎥ ⎢ ⎥ - ⅈ 0 0 0 1 0 4 0 ⎢ ⎥ - 1 0 1 ⎢ ⎥ ⎢ 0 0 0 0 1 0 4 ⎥ ⎢ ⎥ Figure: Γ ˆ F A & ⎣ ⎦ 0 0 0 0 0 1 0 W ( A ) = W ( A 1 ⊕ A 2 ⊕ A 3 ⊕ A 4 ) 32 ( 25 x 2 + 9 y 2 − 2 t 2 ) F A ( x , y , t ) = − t √ √ √ √ ( 25 x 2 ( 2 + 2 ) − 4 t 2 )( 25 x 2 ( 2 − 2 ) − 4 t 2 ) . 2 ) + 9 y 2 ( 2 + 2 ) + 9 y 2 ( 2 − Camenga & Rault (JC & UA ) Generalized Lax Conjecture 4 / 5

Questions? Kristin Camenga camenga@juniata.edu Patrick Rault rault@email.arizona.edu Camenga & Rault (JC & UA ) Generalized Lax Conjecture 5 / 5