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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

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