On the symmetric solutions of a linear matrix equation. (English) Zbl 0622.15001

Let A be a real \(m\times n\)-matrix. An \(n\times m\)-matrix G is called a minimum-norm reflexive generalized inverse (MNRGI) of A, if the following relations hold: (i) \(AGA=A\), (ii) \(GAG=G\), (iii) \(GA=(GA)^ T\). Partitioned MNRGIs are used to derive a necessary and sufficient condition for the consistency of the linear system (*) \(AX=B\), \(X=X^ T\), and to establish the explicit form of the general solution of (*) in that case. Furthermore, the dimension of the solution space to (*) is computed.
Reviewer: A.R.Kräuter


15A09 Theory of matrix inversion and generalized inverses
15A24 Matrix equations and identities
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