On the phases of a complex matrix

By Dan Wang, Wei Chen, Sei Zhen Khong, L. Qiu. Linear Algebra and its Applications, 2019

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In this paper, we define the phases of a complex sectorial matrix to be its canonical angles, which are uniquely determined from a sectorial decomposition of the matrix. Various properties of matrix phases are studied, including those of compressions, Schur complements, matrix products, and sums. In particular, by exploiting a notion known as the compound numerical range, we establish a majorization relation between the phases of the eigenvalues of $AB$ and the phases of $A$ and $B$. This is then applied to investigate the rank of $I + AB$ with phase information of $A$ and $B$, which plays a crucial role in feedback stability analysis. A pair of problems: banded sectorial matrix completion and decomposition is studied. The phases of the Kronecker and Hadamard products are also discussed.

Subfields: Algebra and Number Theory, Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/63a5a03dda4b65055f5a080dff6e1f2a8806e749