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The solvability conditions for the inverse problem of symmetric nonnegative definite matrices. (English. Chinese original) Zbl 0973.15008
Chin. J. Numer. Math. Appl. 12, No. 1, 1-7 (1990); translation from Math. Numer. Sin. 11, No. 4, 337-343 (1989).
Summary: We consider the following two problems:
Problem A: Given \(X,B\in \mathbb{R}^{n\times m}\), find \(A\in P_n\) such that \[ AX=B, \] where \(P_n=\{A\in\mathbb{R}^{n\times n}\mid A=A^T\), \(x^T Ax\geq 0\), \(\forall x\in \mathbb{R}^n\}\).
Problem B: Given \(A^*\in \mathbb{R}^{n\times n}\), find \(\widehat A\in S_E\) such that \[ \|A^*-\widehat A\|= \inf_{A\in S_E}\|A^*-A\|, \] where \(\|\cdot\|\) is Frobenius norm, and \(S_E\) is the solution set of Problem A.
Necessary and sufficient conditions under which \(S_E\) is nonempty are studied. The general form of \(S_E\) is given. For Problem B, the expression of the solution is provided.

15A29 Inverse problems in linear algebra
15A18 Eigenvalues, singular values, and eigenvectors