A kind of inverse problems of matrices and its numerical solution. (Chinese. English summary) Zbl 0641.65037

Zhang, Lei

Math. Numer. Sin. 9, 431-437 (1987).

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)

Keywords:

approximation of a matrix with spectral constraints; inverse problem; numerical stability