The Gau-Wu Number for 4 4 Matrices Kristin A. Camenga, Patrick X. - - PowerPoint PPT Presentation

The Gau-Wu Number for 4 × 4 Matrices

Kristin A. Camenga, Patrick X. Rault, Tsvetanka Sendova, Ilya M. Spitkovsky July 24, 2017

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The Gau-Wu Number, k(A)

Notation: Mn(C): the algebra of all n × n matrices with complex entries ·, ·: the scalar product on Cn ·: the associated norm Definition The numerical range of A ∈ Mn(C): W (A) = {Ax, x: x = 1, x ∈ Cn} .

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The Gau-Wu Number, k(A)

Notation: Mn(C): the algebra of all n × n matrices with complex entries ·, ·: the scalar product on Cn ·: the associated norm Definition The numerical range of A ∈ Mn(C): W (A) = {Ax, x: x = 1, x ∈ Cn} . Definition Let A ∈ Mn(C). We define k(A) to be the maximal size of an orthonormal set {x1, . . . , xk} ⊂ Cn such that the values Axj, xj lie on the boundary ∂W (A) of W (A). We will call k(A) the Gau-Wu number of A.

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Example of k(A)

• 1

2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7

z1 z2 z3 z4

A is a 7 × 7 tridiagonal Toeplitz matrix:          a c . . . b a c ... . . . ... ... ... . . . ... b a c . . . b a          with a = 5 + 4i, b = −1 + i, c = −3. [1]

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Example of k(A)

• 1

2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7

z1 z2 z3 z4

A is a 7 × 7 tridiagonal Toeplitz matrix:          a c . . . b a c ... . . . ... ... ... . . . ... b a c . . . b a          with a = 5 + 4i, b = −1 + i, c = −3. [1] Observe: If A ∈ Mn(C), 2 ≤ k(A) ≤ n ([3, Lemma 4.1]) Wang and Wu have classified the values of k(A) when n = 3.

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The Associated Curve and Boundary Generating Curve

Definition Let A ∈ Mn(C) and denote H1 = (A + A∗) /2 and H2 = (A − A∗) /2i, where A∗ stands for the conjugate transpose of A. Let F(x : y : t) = det (xH1 + yH2 + tIn) , a homogeneous polynomial of degree n. Furthermore, let ΓF denote the curve F(x : y : t) = 0 in projective space CP2. We call F the associated polynomial and ΓF the associated curve for the matrix A. Let Γ∧

F denote

the dual curve of ΓF. We call Γ∧

F the boundary generating curve.

W (A) is the convex hull of the real affine points on Γ∧

F [4, 5]

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The Associated Curve and Boundary Generating Curve

Let A =     1 1 1 1 1 1    . Then the associated curve is 16F(t, x, y) = 16t4 −24t2x2 −24t2y2 +16tx3 +16txy2 −3x4 −2x2y2 +y4 [2, Example 4.1]

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The Associated Curve and Boundary Generating Curve

Let A =     1 1 1 1 1 1    . Then the associated curve is 16F(t, x, y) = 16t4 −24t2x2 −24t2y2 +16tx3 +16txy2 −3x4 −2x2y2 +y4 [2, Example 4.1]

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Flat portions and Maximal k(A)

For irreducible matrices, flat portions correspond to a singularity on ΓF. The order of the singularity is the multiplicity of the corresponding eigenvalue of H1.

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Flat portions and Maximal k(A)

For irreducible matrices, flat portions correspond to a singularity on ΓF. The order of the singularity is the multiplicity of the corresponding eigenvalue of H1. Proposition Let A ∈ Mn(C) be an irreducible matrix with associated curve ΓF. k(A) = n if and only if ΓF has two real points P and Q which are collinear with the origin and satisfy ordP (ΓF) + ordQ (ΓF) = n.

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More examples of maximal k(A) for n = 4

[2, Example 4.7] [2, Example 4.2]

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k(A) = 2

If A is irreducible and k(A) = 2, there can be no flat portion. Let A =     1

1 2

2    . This is irreducible and k(A) = 2 by [6, Theorems 3.2, 3.10]. The associated curve and boundary generating curve are

• 10
• 5

5 10

• 10
• 5

5 10

• 1

1 2

• 1

1

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4 × 4 A with k(A) = 3

• 1

1

• 2
• 1

1 2



• 2
• 1

1 2

• 2
• 1

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Non-singular 4 × 4 A for which k(A) = 3

Suppose k(A) = 3, where A is irreducible and ΓF 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:

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Case 1: no parallel supporting lines

Suppose e1, e2, e3 map to the imaginary axis, real axis, and x + y = k respectively. If we write A = H + iK for Hermitian H and K, then H =     h22 h24 h33 h34 h42 h43 h44     and K =     k11 k14 k33 k34 k41 k43 k44    . where k34 = −h34 and k43 = −h43 the 1,1 minor of H and the 2,2 minor of K are positive definite h33 + k33 is bigger than the eigenvalues of the 3,3 minor of H + K

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Case 1 Example

A =     2i

3i 8

1

1 8 1 4 + i 2 1 8 − i 8 3i 8 1 8 1 8 − i 8

1 + i    

        1 1 2

The boundary generating curve and numerical range for A. Note that in this case A has the form of an arrowhead matrix.

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Case 2 Example

For this case, if A = H + iK, the Hermitian H and K take the forms H =     1 h33 h34 h43 h44     and K =     k11 k12 k14 k21 k22 k24 k41 k42 k44    . A =     2i i 1 + 2i i

3 4 1 4

i i

1 4 1 2 + 3i

   

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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.

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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

Sn-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 (2013), no. 1, 514–532.

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