  Let $F(x) := (f_{ij}(x))_{i,j=1,\ldots,p},$ be a real symmetric polynomial
matrix of order $p$ and let $f(x)$ be the largest eigenvalue function of the
matrix $F(x).$ We denote by ${\partial}^\circ f(x)$ the Clarke subdifferential
of $f$ at $x.$ In this paper, we first give the following {\em nonsmooth}
version of \L ojasiewicz gradient inequality for the function $f$ with an
explicit exponent: For any $\bar x\in \Bbb R^n$ there exist $c > 0$ and
$\epsilon > 0$ such that we have for all $\|x - \bar{x}\| < \epsilon,$
\begin{equation*} \inf \{ \| w \| \ : \ w \in {\partial}^\circ f(x) \} \ \ge \
c\, |f(x) - f(\bar x)|^{1 - \frac{1}{\mathscr{R}(2n+p(n+1),d+3)}},
\end{equation*} where $d:=\max_{i,j = 1, \ldots, p}\deg f_{i j}$ and
$\mathscr{R}$ is a function introduced by D'Acunto and Kurdyka: $\mathscr{R}(n,
d) := d(3d - 3)^{n-1}$ if $d \ge 2$ and $\mathscr{R}(n, d) := 1$ if $d = 1.$
Then we establish error bounds with explicitly determined exponents, local and
global, for the largest eigenvalue function $f(x)$ of the matrix $F(x)$.
