Diagonal Norm Hermitian Matrices
نویسنده
چکیده
If v is a norm on en, let H(v) denote the set of all norm-Hermitians in e nn. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S = H(v) (or S = H(v) n D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues .111" '" AT' r ~ n, there is a norm v such that hE H(v) , but h ¢= H(v), for some integer s, if and only if .112 .Ill" .. , AT .Ill are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms.
منابع مشابه
Departure from Normality and Eigenvalue Perturbation Bounds
Perturbation bounds for eigenvalues of diagonalizable matrices are derived that do not depend on any quantities associated with the perturbed matrix; in particular the perturbed matrix can be defective. Furthermore, Gerschgorin-like inclusion regions in the Frobenius are derived, as well as bounds on the departure from normality. 1. Introduction. The results in this paper are based on two eigen...
متن کاملCartesian decomposition of matrices and some norm inequalities
Let X be an n-square complex matrix with the Cartesian decomposition X = A + i B, where A and B are n times n Hermitian matrices. It is known that $Vert X Vert_p^2 leq 2(Vert A Vert_p^2 + Vert B Vert_p^2)$, where $p geq 2$ and $Vert . Vert_p$ is the Schatten p-norm. In this paper, this inequality and some of its improvements ...
متن کاملAsymptotic quadratic convergence of the serial block-Jacobi EVD algorithm for Hermitian matrices
This report is devoted to the proof of the local (asymptotic) quadratic convergence of the serial block-Jacobi EVD algorithm for Hermitian matrices with multiple eigenvalues. At each iteration step, two off-diagonal blocks with the largest Frobenius norm are annihilated which is an extension of the ‘classical’ Jacobi approach to the block case.
متن کاملAn iterative method for the Hermitian-generalized Hamiltonian solutions to the inverse problem AX=B with a submatrix constraint
In this paper, an iterative method is proposed for solving the matrix inverse problem $AX=B$ for Hermitian-generalized Hamiltonian matrices with a submatrix constraint. By this iterative method, for any initial matrix $A_0$, a solution $A^*$ can be obtained in finite iteration steps in the absence of roundoff errors, and the solution with least norm can be obtained by choosing a special kind of...
متن کاملThe Distance between the Eigenvalues of Hermitian Matrices
It is shown that the minmax principle of Ky Fan leads to a quick simple derivation of a recent inequality of V. S. Sunder giving a lower bound for the spectral distance between two Hermitian matrices. This brings out a striking parallel between this result and an earlier known upper bound for the spectral distance due to L. Mirsky. Let A be a Hermitian matrix of order n and let A ¿(A) denote th...
متن کامل