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Register a new userThe Gau-Wu Number for $4times 4$ and Select Arrowhead Matrices
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The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau--Wu number (i.e., the maximal number $k(A)$ of orthonormal vectors $x_j$ such that the scalar products $langle Ax_j,x_jrangle$ lie on the boundary of the numerical range of $A$) is computed for a class of arrowhead matrices $A$ of arbitrary size, including dichotomous ones. These results are then used to completely classify all $4times4$ matrices according to the values of their Gau--Wu numbers.
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In 2013, Gau and Wu introduced a unitary invariant, denoted by $k(A)$, of an $ntimes n$ matrix $A$, which counts the maximal number of orthonormal vectors $textbf x_j$ such that the scalar products $langle Atextbf x_j,textbf x_jrangle$ lie on the boundary of the numerical range $W(A)$. We refer to $k(A)$ as the Gau--Wu number of the matrix $A$. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify $k(A)$. This continues the work of Wang and Wu classifying the Gau-Wu numbers for $3times 3$ matrices. Our focus on singularities is inspired by Chien and Nakazato, who classified $W(A)$ for $4times 4$ unitarily irreducible $A$ with irreducible base curve according to singularities of that curve. When $A$ is a unitarily irreducible $ntimes n$ matrix, we give necessary conditions for $k(A) = 2$, characterize $k(A) = n$, and apply these results to the case of unitarily irreducible $4times 4$ matrices. However, we show that knowledge of the singularities is not sufficient to determine $k(A)$ by giving examples of unitarily irreducible matrices whose base curves have the same types of singularities but different $k(A)$. In addition, we extend Chien and Nakazatos classification to consider unitarily irreducible $A$ with reducible base curve and show that we can find corresponding matrices with identical base curve but different $k(A)$. Finally, we use the recently-proved Lax Conjecture to give a new proof of a theorem of Helton and Spitkovsky, generalizing their result in the process.
Gau, Wang and Wu in their LAMA2016 paper conjectured (and proved for $nleq 4$) that an $n$-by-$n$ partial isometry cannot have a circular numerical range with a non-zero center. We prove that this statement holds also for $n=5$.
A complete description of 4-by-4 matrices $begin{bmatrix}alpha I & C D & beta Iend{bmatrix}$, with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which $C-D^*$ also is a scalar multiple of the identity. In particular cases when in addition $alpha-beta$ is real or pure imaginary, the results take an especially simple form. An application to reciprocal matrices is provided.
The 4-by-4 nilpotent matrices the numerical ranges of which have non-parallel flat portions on their boundary that are on lines equidistant from the origin are characterized. Their numerical ranges are always symmetric about a line through the origin and all possible angles between the lines containing the flat portions are attained.
The product of a Hermitian matrix and a positive semidefinite matrix has only real eigenvalues. We present bounds for sums of eigenvalues of such a product.
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