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A kind of inverse problems of matrices and its numerical solution. (Chinese. English summary) Zbl 0641.65037
This paper discusses the following inverse problem. Given $$C\in R^{n\times n}$$, find $$\hat A\in S$$ such that $$\| C-\hat A\| =\inf_{\forall A\in S}\| C-A\|,$$ where $$\| \cdot \|$$ is the Frobenius norm and S denotes the solution set of the following problem. Given $$X,B\in R^{n\times m}$$, find $$A\in P_{\alpha}$$ such that $$AX=B$$, where $$P_{\alpha}=\{A\in R^{n\times m}|^ A=A^ T,\quad X^ TAX\geq \alpha X^ Tx,\quad \alpha \geq 0,\quad \forall X\in R^ n\}.$$ The problem for $$\alpha =0$$ is discussed in detail. Necessary condition and some sufficient conditions under which S is nonempty are studied and the general form of S is given. The solution of the problem is analyzed and a representation of the solution $$\hat A$$ is given. Finally, an algorithm for solving the problem is presented and the numerical stability of the algorithm is pointed out.
Reviewer: Xie Shenquan

##### MSC:
 65F30 Other matrix algorithms (MSC2010)