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Inertias and ranks of some Hermitian matrix functions with applications. (English) Zbl 1253.15050

The paper deals with the following three related topics involving Hermitian matrices: (a) an optimization problem for a matrix function with linear constraints, (b) solvability of certain systems of linear equations, (c) properties of the range of values of certain matrix functions of positive semidefinite matrices. The motivation to study these topics comes from their relevance for control and stability theory and, in particular, their relationship with the discrete Lyapunov equation.
Let \(A\) be a complex matrix (not necessarily square) and \(H\) be a Hermitian matrix. The technical means used in the paper involve in particular the rank \(r(A)\) of matrix \(A\), its Moore-Penrose pseudoinverse, and the positive and the negative index \(i_\pm(H)\) of inertia of \(H\), i.e., the number of positive/negative eigenvalues.
First, the maximum and the minimum values of \(r\) and \(i_\pm\) are characterized over the range of values of the matrix function \[ f(X,Y)=P-QXQ^*-TYT^*, \] where \(P,Q,T\) are given matrices, \(P\) is Hermitian, and the asterisk denotes the conjugate transpose. The arguments \(X,Y\) are Hermitian matrices satisfying the constraints \[ AX=B,\qquad YC=D \] for given matrices \(A,B\) and \(C,D\).
This result is then used in topic (a) to express conditions under which the function \(f\) can be maximized in the following sense: \(f\) attains its maximum at \((X_0,Y_0)\) if for any admissible \((X,Y)\) the matrix \(f(X_0,Y_0)-f(X,Y)\) is positive semidefinite. In this case it follows that the maximum value of \(f\) does not depend on the particular maximizing pair \((X_0,Y_0)\).
Next, in topic (b), the solvability conditions of \(f(X,Y)=0\) under the above constraints are specified and the solutions are expressed explicitly. Some corollaries of this result are considered.
Finally, in topic (c), the maximum and minimum values of \(i_\pm\) over the range of values of the above function \(f\) and function \(g(X,Y)=P+QXQ^*+TYT^*\), respectively, are characterized in the case that the above constraint is replaced by the requirement that \(X,Y\) be both positive semidefinite.
The paper is written in a clear manner. The exposition is complemented by several examples. The terms “positive semidefinite” and “nonnegative definite” are used interchangeably.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A03 Vector spaces, linear dependence, rank, lineability
15A09 Theory of matrix inversion and generalized inverses
15A24 Matrix equations and identities
15B48 Positive matrices and their generalizations; cones of matrices
65F30 Other matrix algorithms (MSC2010)
15A18 Eigenvalues, singular values, and eigenvectors
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