  Let $\bm{x}_1,\cdots,\bm{x}_n$ be a random sample of size $n$ from a
$p$-dimensional population distribution, where $p=p(n)\rightarrow\infty$.
Consider a symmetric matrix $W=X^\top X$ with parameters $n$ and $p$, where
$X=(\bm{x}_1,\cdots,\bm{x}_n)^\top$. In this paper, motivated by model
selection theory in high-dimensional statistics, we mainly investigate the
asymptotic behavior of the eigenvalues of the principal minors of the random
matrix $W$. For the Gaussian case, under a simple condition that $m=o(n/\log
p)$, we obtain the asymptotic results on maxima and minima of the eigenvalues
of all $m\times m$ principal minors of $W$. We also extend our results to
general distributions with some moment conditions. Moreover, we gain the
asymptotic results of the extreme eigenvalues of the principal minors in the
case of the real Wigner matrix. Finally, similar results for the maxima and
minima of the eigenvalues of all the principal minors with a size smaller than
or equal to $m$ are also given.
