@Article{NM-15-227, author = {X. Pan, X. Hu and L. Zhang}, title = {A class of constrained inverse eigenproblem and associated approximation problem for symmetric reflexive matrices}, journal = {Numerical Mathematics, a Journal of Chinese Universities}, year = {2006}, volume = {15}, number = {3}, pages = {227--236}, abstract = { Let $S\in R^{n\times n}$ be a symmetric and nontrival involution matrix. We say that $A\in R^{n\times n}$ is a symmetric reflexive matrix if $A^T=A$ and $SAS=A$. Let $SR^{n\times n}_r(S)$=\{$A|A=A^T, A=SAS, A\in R^{n\times n}$\}. This paper discusses the following two problems. The first one is as follows. Given $Z\in R^{n\times m}$ $(m<n)$, $\Lambda= diag(\lambda_{_1},\cdots, \lambda_{_m})\in R^{m\times m}$, and $\alpha, \beta\in R$ with $\alpha <\beta$. Find a subset $\varphi(Z,\Lambda,\alpha,\beta)$ of $SR^{n\times n}_r(S)$ such that $AZ=Z\Lambda$ holds for any $A\in \varphi(Z,\Lambda,\alpha,\beta)$ and the remaining eigenvalues $\lambda_{_{m+1}},\cdots,\lambda_{_n}$ of $A$ are located in the interval $[\alpha,\beta]$. Moreover, for a given $B\in R^{n\times n}$, the second problem is to find $A_B\in \varphi(Z,\Lambda,\alpha,\beta)$ such that $$ \|B-A_B\|=\min_{A\in \varphi(Z,\Lambda,\alpha,\beta)}\|B-A\|, $$ where $\|.\|$ is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived. }, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/nm/8030.html} }