The Gau-Wu Number for 4 × 4 Matrices Kristin A. Camenga, Patrick X. Rault, Tsvetanka Sendova, Ilya M. Spitkovsky July 24, 2017 Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 1 / 15
The Gau-Wu Number, k ( A ) Notation: M n ( C ): the algebra of all n × n matrices with complex entries �· , ·� : the scalar product on C n �·� : the associated norm Definition The numerical range of A ∈ M n ( C ) : W ( A ) = {� A x , x � : � x � = 1 , x ∈ C n } . Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 2 / 15
The Gau-Wu Number, k ( A ) Notation: M n ( C ): the algebra of all n × n matrices with complex entries �· , ·� : the scalar product on C n �·� : the associated norm Definition The numerical range of A ∈ M n ( C ) : W ( A ) = {� A x , x � : � x � = 1 , x ∈ C n } . Definition Let A ∈ M n ( C ) . We define k ( A ) to be the maximal size of an orthonormal set { x 1 , . . . , x k } ⊂ C n such that the values � A x j , x j � lie on the boundary ∂ W ( A ) of W ( A ) . We will call k ( A ) the Gau-Wu number of A . Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 2 / 15
Example of k ( A ) A is a 7 × 7 tridiagonal 7 z 3 Toeplitz matrix: 6 � 0 . . . 0 a c 5 . ... � . . b a c z 2 z 4 4 ... ... ... 0 0 3 � . ... . . b a c � 2 z 1 0 . . . 0 b a 1 with a = 5 + 4 i , b = − 1 + i , c = − 3. [1] 1 2 3 4 5 6 7 8 9 10 Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 3 / 15
Example of k ( A ) A is a 7 × 7 tridiagonal 7 z 3 Toeplitz matrix: 6 � 0 . . . 0 a c 5 . ... � . . b a c z 2 z 4 4 ... ... ... 0 0 3 � . ... . . b a c � 2 z 1 0 . . . 0 b a 1 with a = 5 + 4 i , b = − 1 + i , c = − 3. [1] 1 2 3 4 5 6 7 8 9 10 Observe: If A ∈ M n ( C ), 2 ≤ k ( A ) ≤ n ([3, Lemma 4.1]) Wang and Wu have classified the values of k ( A ) when n = 3. Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 3 / 15
The Associated Curve and Boundary Generating Curve Definition Let A ∈ M n ( C ) and denote H 1 = ( A + A ∗ ) / 2 and H 2 = ( A − A ∗ ) / 2 i , where A ∗ stands for the conjugate transpose of A. Let F ( x : y : t ) = det ( xH 1 + yH 2 + tI n ) , a homogeneous polynomial of degree n. Furthermore, let Γ F denote the curve F ( x : y : t ) = 0 in projective space CP 2 . We call F the associated polynomial and Γ F the associated curve for the matrix A. Let Γ ∧ F denote the dual curve of Γ F . We call Γ ∧ F the boundary generating curve . W ( A ) is the convex hull of the real affine points on Γ ∧ F [4, 5] Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 4 / 15
The Associated Curve and Boundary Generating Curve 0 1 1 1 0 0 1 1 Let A = . Then the associated curve is 0 0 0 1 0 0 0 0 16 F ( t , x , y ) = 16 t 4 − 24 t 2 x 2 − 24 t 2 y 2 +16 tx 3 +16 txy 2 − 3 x 4 − 2 x 2 y 2 + y 4 [2, Example 4.1] Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 5 / 15
The Associated Curve and Boundary Generating Curve 0 1 1 1 0 0 1 1 Let A = . Then the associated curve is 0 0 0 1 0 0 0 0 16 F ( t , x , y ) = 16 t 4 − 24 t 2 x 2 − 24 t 2 y 2 +16 tx 3 +16 txy 2 − 3 x 4 − 2 x 2 y 2 + y 4 [2, Example 4.1] Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 5 / 15
Flat portions and Maximal k ( A ) For irreducible matrices, flat portions correspond to a singularity on Γ F . The order of the singularity is the multiplicity of the corresponding eigenvalue of H 1 . Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 6 / 15
Flat portions and Maximal k ( A ) For irreducible matrices, flat portions correspond to a singularity on Γ F . The order of the singularity is the multiplicity of the corresponding eigenvalue of H 1 . Proposition Let A ∈ M n ( C ) be an irreducible matrix with associated curve Γ F . k ( A ) = n if and only if Γ F has two real points P and Q which are collinear with the origin and satisfy ord P (Γ F ) + ord Q (Γ F ) = n. Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 6 / 15
More examples of maximal k ( A ) for n = 4 [2, Example 4.7] [2, Example 4.2] Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 7 / 15
k ( A ) = 2 If A is irreducible and k ( A ) = 2, there can be no flat portion. 0 1 0 0 1 0 0 0 2 Let A = . This is irreducible and k ( A ) = 2 by [6, Theorems 0 0 0 2 0 0 0 0 3.2, 3.10]. The associated curve and boundary generating curve are 10 1 5 0 0 - 5 - 1 - 10 - 10 - 5 0 5 10 - 1 0 1 2 Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 8 / 15
4 × 4 A with k ( A ) = 3 2 2 1 1 0 0 - 1 - 1 - 2 - 2 - 2 - 1 0 1 2 - 1 0 1 Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 9 / 15
Non-singular 4 × 4 A for which k ( A ) = 3 Suppose k ( A ) = 3, where A is irreducible and Γ F is non-singular. Consider a maximum orthogonal set that maps to the boundary and all the supporting lines at the corresponding image points on the boundary. No two of the supporting lines are parallel Two of the supporting lines are parallel and the third point is between Via affine transformations we can transform W ( A ) in these two to a convex shape inscribed in one of the following: Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 10 / 15
Case 1: no parallel supporting lines Suppose e 1 , e 2 , e 3 map to the imaginary axis, real axis, and x + y = k respectively. If we write A = H + iK for Hermitian H and K , then 0 0 0 0 k 11 0 0 k 14 0 h 22 0 h 24 0 0 0 0 H = and K = . 0 0 h 33 h 34 0 0 k 33 k 34 0 h 42 h 43 h 44 k 41 0 k 43 k 44 where k 34 = − h 34 and k 43 = − h 43 the 1,1 minor of H and the 2,2 minor of K are positive definite h 33 + k 33 is bigger than the eigenvalues of the 3,3 minor of H + K Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 11 / 15
Case 1 Example 3 i 2 i 0 0 8 1 0 1 0 A = 8 1 1 4 + i 8 − i 0 0 2 8 3 i 1 1 8 − i 1 + i 8 8 8 2 1 0 0 1 The boundary generating curve and numerical range for A . Note that in this case A has the form of an arrowhead matrix. Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 12 / 15
Case 2 Example For this case, if A = H + iK , the Hermitian H and K take the forms 0 0 0 0 0 k 11 k 12 k 14 0 1 0 0 k 21 k 22 0 k 24 H = and K = . 0 0 0 0 0 0 h 33 h 34 0 0 h 43 h 44 k 41 k 42 0 k 44 2 i 0 0 i 0 1 + 2 i 0 i A = 3 1 0 0 4 4 1 1 i i 2 + 3 i 4 Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 13 / 15
Summary Singularities and their locations on the associated curve characterize maximal k ( A ) for all n . For k ( A ) = 2, the associated curve has no singularities. k ( A ) can be greater than 2 even if there are no singularities on the associated curve. To classify k ( A ) for 4 × 4 matrices, the challenge is differentiating between k ( A ) = 2 and k ( A ) = 3. Camenga, Rault, Sendova, Spitkovsky 7/24/17 July 24, 2017 14 / 15
References K. Camenga, P. X. Rault, T. Sendova, and I. M. Spitkovsky, On the Gau–Wu number for some classes of matrices , Linear Algebra and its Applications 444 (2014), 254–262. M. T. Chien and H. Nakazato, Singular points of the ternary polynomials associated with 4-by-4 matrices , Electronic Journal of Linear Algebra 23 (2012), 755–769. H. L. Gau and P. Y. Wu, Numerical ranges and compressions of S n -matrices , Operators and Matrices 7 (2013), no. 2, 465–476. R. Kippenhahn, ¨ Uber den Wertevorrat einer Matrix , Mathematische Nachrichten 6 (1951), 193–228. , On the numerical range of a matrix , Linear and Multilinear Algebra 56 (2008), no. 1-2, 185–225, Translated from the German by Paul F. Zachlin and Michiel E. Hochstenbach K. Z. Wang and P. Y. Wu, Diagonals and numerical ranges of weighted shift matrices , Linear Algebra and its Applications 438 Camenga, Rault, Sendova, Spitkovsky (2013), no. 1, 514–532. 7/24/17 July 24, 2017 15 / 15
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