Some counterexamples related to sectorial matrices and matrix phases

Abstract A sectorial matrix is an n × n matrix whose numerical range is contained in an open half-plane, and such matrices have many nice properties. In particular, the subset of strictly accretive matrices is a convex cone in the space of n × n matrices, and results related to positive definite matrices have recently been generalized to this cone. Moreover, sectorial matrices have recently been used to define phases of a matrix, and these phases can be used to angularly bound the eigenvalues by majorization-type inequalities similar to the ones for the singular values and the absolute value of the eigenvalues. Nevertheless, many traits that would be desirable are not true for sectorial matrices and matrix phases, and in this note we present a number of counterexamples for such traits. More precisely, the counterexamples are related to sectorial polar decompositions, majorization inequalities for phases of products, the spectral and numerical radius, and Schur complements.

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