Ela Spectral Properties of Sign Symmetric Matrices
نویسنده
چکیده
Spectral properties of sign symmetric matrices are studied. A criterion for sign symmetry of shifted basic circulant permutation matrices is proven, and is then used to answer the question which complex numbers can serve as eigenvalues of sign symmetric 3 × 3 matrices. The results are applied in the discussion of the eigenvalues of QM -matrices. In particular, it is shown that for every positive integer n there exists a QM -matrix A such that Ak is a sign symmetric P -matrix for all k ≤ n, but not all the eigenvalues of A are positive real numbers.
منابع مشابه
Properties of eigenvalue function
For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties.We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.
متن کاملEla Multiplicative Maps on Invertible Matrices That Preserve Matricial Properties
Descriptions are given of multiplicative maps on complex and real matrices that leave invariant a certain function, property, or set of matrices: norms, spectrum, spectral radius, elementary symmetric functions of eigenvalues, certain functions of singular values, (p, q) numerical ranges and radii, sets of unitary, normal, or Hermitian matrices, as well as sets of Hermitian matrices with fixed ...
متن کاملEla Variational Characterizations of the Sign-real and the Sign-complex Spectral Radius∗
The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the classical Perron-Frobenius theory (for nonnegative matrices) to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one cor...
متن کاملEla Sign Patterns of the Schwarz Matrices and Generalized Hurwitz Polynomials
The direct and inverse spectral problems are solved for a wide subclass of the class of Schwarz matrices. A connection between Schwarz matrices and the so-called generalized Hurwitz polynomials is found. The known results due to H. Wall and O. Holtz are briefly reviewed and obtained as particular cases.
متن کاملProperties of Central Symmetric X-Form Matrices
In this paper we introduce a special form of symmetric matrices that is called central symmetric $X$-form matrix and study some properties, the inverse eigenvalue problem and inverse singular value problem for these matrices.
متن کامل