Max-min optimizations on the rank and inertia of a linear Hermitian matrix expression subject to range, rank and definiteness restrictions
نویسنده
چکیده
The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we give various closed-form formulas for the maximal and minimal values for the rank and inertia of the Hermitian expression A + X, where A is a given Hermitian matrix and X is a variable Hermitian matrix satisfying the range and rank restrictions range(X) ⊆ range(B) and rank(X) 6 k. Some expressions of the Hermitian matrix X such that A + X attains the extremal ranks and inertias are also presented.
منابع مشابه
Investigation on the Hermitian matrix expression subject to some consistent equations
In this paper, we study the extremal ranks and inertias of the Hermitian matrix expression $$ f(X,Y)=C_{4}-B_{4}Y-(B_{4}Y)^{*}-A_{4}XA_{4}^{*},$$ where $C_{4}$ is Hermitian, $*$ denotes the conjugate transpose, $X$ and $Y$ satisfy the following consistent system of matrix equations $A_{3}Y=C_{3}, A_{1}X=C_{1},XB_{1}=D_{1},A_{2}XA_{2}^{*}=C_{2},X=X^{*}.$ As consequences, we g...
متن کاملAnalytical formulas for calculating the extremal ranks and inertias of A + BXB∗ when X is a fixed-rank Hermitian matrix
The rank of a matrix and the inertia of a square matrix are two of the most generic concepts in matrix theory for describing the dimension of the row/column vector space and the sign distribution of the eigenvalues of the matrix. Matrix rank and inertia optimization problems are a class of discontinuous optimization problems, in which decision variables are matrices running over certain matrix ...
متن کاملOn global optimizations of the rank and inertia of the matrix function
For a given linear matrix function A1−B1XB 1 , where X is a variable Hermitian matrix, this paper derives a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the matrix function subject to a pair of consistent matrix equations B2XB ∗ 2 = A2 and B3XB ∗ 3 = A3. As applications, we give necessary and sufficient conditions for the triple matrix equat...
متن کاملEla Rank and Inertia of Submatrices of the Moore–penrose Inverse of a Hermitian Matrix
Closed-form formulas are derived for the rank and inertia of submatrices of the Moore–Penrose inverse of a Hermitian matrix. A variety of consequences on the nonsingularity, nullity and definiteness of the submatrices are also presented.
متن کاملRank and inertia optimizations of two Hermitian quadratic matrix functions subject to restrictions with applications
In this paper, we first give the maximal and minimal values of the ranks and inertias of the quadratic matrix functions q1(X) = Q1 − XP1X and q2(X) = Q2 − XP2X subject to a consistent matrix equation AX = B, where Q1, Q2, P1 and P2 are Hermitian matrices. As applications, we derive necessary and sufficient conditions for the solution of AX = B to satisfy the quadratic equality XP1X ∗ = Q1 and X...
متن کامل