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Matrix quadratic solutions. (English) Zbl 0144.02001
Let \(A\), \(B\), \(C\), and \(X\) be \(n\times n\) matrices with complex entries. In case \(M = \begin{bmatrix} B &A \\ C &-B^*\end{bmatrix}\) has diagonal Jordan canonical form, the author obtains a representation for solutions of
\[ A + BX + XB^* - XCX = 0 \tag{1} \]
in terms of eigenvectors of \(M\).
Theorem. Let \(M\) have diagonal Jordan canonical form. Every solution of (1) has the form \([b_1,\ldots, b_n]\,[c_1, \ldots, c_n]^{-1}\) where \(b_n\), \(c_n\) are column vectors and \(\begin{bmatrix} b_i \\ c_i\end{bmatrix}\) is an eigenvalue of \(M\). Conversely, if \(\begin{bmatrix} b_i \\ c_i\end{bmatrix}\) is an eigenvector of \(M\) \((1\le i\le n)\) and \([c_1, \ldots, c_n]\) is non-singular, then \([b_1,\ldots, b_n]\,[c_1, \ldots, c_n]^{-1}\) is a solution of (1).
We note here the first part does not necessarily hold when \(M\) is not diagonalizable, but the converse is valid without the restriction on \(M\).
In two additional theorems the author obtains sufficient conditions that (1) have Hermitian and positive definite solutions.
Reviewer: Irving J. Katz

15A24 Matrix equations and identities
15A18 Eigenvalues, singular values, and eigenvectors
15A20 Diagonalization, Jordan forms
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