From real to complex sign pattern matrices

Abstract

This paper extends some fundamental concepts of qualitative matrix analysis from sign pattern classes of real matrices to sign pattern classes of complex matrices. A complex sign pattern and its corresponding sign pattern class are defined in such a way that they generalize the definitions of a (real) sign pattern and its corresponding sign pattern class. A survey of several qualitative results on complex sign patterns is presented. In particular, sign nonsingular complex patterns are investigated. The type of region in the complex plane representing the distribution of the determinants of the matrices in the sign pattern class of a sign nonsingular complex pattern is identified. Cyclically nonnegative complex patterns and complex patterns that are signature similar to nonnegative patterns are characterized. Extensions of sign stable and sign semistable patterns from the real to the complex case are given. Results on ray patterns are also obtained. Finally, many open questions are mentioned.

Eigenvalue distribution of certain ray patterns

“…This characterization shows that cyclically real ray patterns generalize real sign patterns in the sense that the eigenvalue distribution of any irreducible cyclically real pattern is the same as the eigenvalue distribution of a real sign pattern. This is analogous to the generalization of nonnegative patterns to cyclically nonnegative patterns, see [3]. Cyclically real ray patterns A are also characterized in terms of the spectra of the matrices in R(A).…”

“…In particular, we characterize the patterns that require all real eigenvalues, and the patterns that require all pure imaginary eigenvalues. In Section 4, we discuss the more general sector patterns (see [3]), and we give some open questions concerning ray patterns and sector patterns.…”

“…Recall that a matrix B is said to be stable if each eigenvalue of B has negative real part, that is, lies in the left half-plane. As in [3], we say a ray pattern A is ray stable if every B ∈ R(A) is stable. The following result is Theorem 5.2 in [3].…”

“…Hence, it is useful to investigate properties of complex matrices based upon the patterns of their entries. Two recent papers ( [3] and [8]) have begun an investigation of patterns of complex matrices.…”

“…It should be clear from the context that e iα means either the ray e iα or the complex number e iα . A ray pattern A = (a kj ) is a matrix, each of whose entries is either 0 or a ray e iα kj (see [3] or [8]). It can be seen from this definition that ray pattern matrices are generalizations of real sign pattern matrices.…”