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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.

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