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A note on the spectral radius of a product of companion matrices
No full textConditions are given on the coefficients of the characteristic polynomials of a set of k companion matrices to ensure that the spectral radius of their product is bounded by tk where 0 < t < 1
Ranks and eigenvalues of states with prescribed reduced states
No full textFor a quantum state represented as an density matrix , let \cS(\sigma) be the compact convex set of quantum states with the first partial trace equal to , i.e., \tr_1(\rho) =\rho_{11} + \cdots + \rho_{mm} = \sigma. It is known that if then there is a rank one matrix \rho \in \cS(\sigma) satisfying \tr_1(\rho) = \sigma. If , there may not be any rank one matrix in \cS(\sigma). In this paper, we determine the ranks of the elements and ranks of the extreme points of the set \cS. We also determine \rho^* \in \cS(\sigma) with rank bounded by such that \|\tr_1(\rho^*) - \sigma\| is minimum for a given unitary similarity invariant norm . Furthermore, the relation between the eigenvalues of and those of \rho \in \cS(\sigma) is analyzed. Extension of the results and open problems will be mentioned
Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations
No full textThe set of all solutions to the homogeneous system of matrix equations (XTA +AX,XTB + BX) = (0, 0), where (A,B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A,B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. K°agstr¨om, and V.V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375–3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils
Submatrices of Hadamard matrices: complementation results
No full textTwo submatrices A, D of a Hadamard matrix H are called complementary if, up to a permutation of rows and columns, H = . In this paper, an explicit formula for the polar decomposition of D is found. As an application, it is shown that under suitable smallness assumptions on the size of A, the complementary matrix D is an almost Hadamard sign pattern, i.e., its rescaled polar part is an almost Hadamard matrix
Possible numbers of nonzero entries in a matrix with a given term rank
No full textThe possible numbers of nonzero entries in a matrix with a given term rank aredetermined respectively in the generic case, the symmetric case and the symmetric case with 0’s on the main diagonal. The matrices that attain the largest number of nonzero entries are also determined
Graded triangular algebras
No full textThe structure of graded triangular algebras T of arbitrary dimension are studied in this paper. This is motivated in part for the important role that triangular algebras play in the study of oriented graphs, upper triangular matrix algebras or nest algebras. It is shown that T decomposes as T = U + (∑i∈I Ti), where U is an R-submodule contained in the 0-homogeneous component and any Ti a well-described (graded) ideal satisfying TiTj = 0 if i≠j. Since any T is not simple asassociative algebra, the concept of quasi-simple triangular algebra is introduced as those T which are as near to simplicity as possible. Under mild conditions, the quasi-simplicity of T is characterized and it is proven that T is the direct sum of quasi-simple graded triangular algebras which are also ideals
