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The Generalized Saddle-Node Bifurcation of Degenerate Solution

Ping Liu;  Yu-Wen Wang — 2005

Commentationes Mathematicae

In this paper we discuss the bifurcation problem for the abstract operator equation of the form F ( u , λ ) = θ with a parameter λ , where F : X × R → Y is a C 1 mapping, X , Y are Banach spaces. By the bounded linear generalized inverse A + of A = F u ( u 0 , λ 0 ) , an abstract bifurcation theorem for the case dim N ( F u ( u 0 , λ 0 ) ) ≥ codim R ( F u ( u 0 , λ 0 ) ) = 1 has been obtained.

The Re-nonnegative definite solutions to the matrix equation A X B = C

Qing Wen Wang;  Chang Lan Yang — 1998

Commentationes Mathematicae Universitatis Carolinae

An n × n complex matrix A is called Re-nonnegative definite (Re-nnd) if the real part of x * A x is nonnegative for every complex n -vector x . In this paper criteria for a partitioned matrix to be Re-nnd are given. A necessary and sufficient condition for the existence of and an expression for the Re-nnd solutions of the matrix equation A X B = C are presented.

Inertias and ranks of some Hermitian matrix functions with applications

Xiang Zhang;  Qing-Wen Wang;  Xin Liu — 2012

Open Mathematics

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications,...

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