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.
