Complementarity forms of theorems of Lyapunov and Stein, and related results
A theorem due to Lyapunov says that if \(A\) is a complex square matrix and if \(K\) is positive definite, then \(A\) is positive stable (i.e. all its eigenvalues lie in the open right half-plane) if and only if there exists a Hermitian positive definite matrix \(X\) such that \(XA+A^*X=K\). The paper gives a complementarity form of this theorem: \(A\) is positive stable if and only if for each Hermitian matrix \(Q\) there exists a Hermitian positive semidefinite matrix \(X\) such that \(AX+XA^*+Q\) is positive semidefinite and \(X[AX+XA^*+Q]=0\). It is shown in the paper that positive stability of \(A\) is equivalent to the so-called \(P_1\)-property of the linear transformation \(L_A(X):=AX+XA^*\): \(X\) Hermitian, \(XL_A(X)+L_A(X)X\) negative semidefinite implies \(X=0\). (A similar property, called \(P\)-property, is known to hold for real matrices \(A\).) Recall Stein's theorem: Let \(A\) be a complex square matrix and \(K\) a Hermitian positive definite one. Then all eigenvalues of \(A\) lie in the open unit disk if and only if there exists a Hermitian positive definite matrix \(X\) such that \(X-AXA^*=K\). The paper gives the complementarity form of this theorem: \(A\) has all its eigenvalues in the open unit disk if and only if for each Hermitian matrix \(Q\) there exists a Hermitian positive semidefinite matrix \(X\) such that \(X-AXA^*+Q\) is positive semidefinite and \(X[X-AXA^*+Q]=0\).
scientific article; zbMATH DE number 1556777
scientific article