  By the classical Sturm's theorem, the number of distinct real roots of a
given real polynomial $f(x)$ within any interval $(a,b]$ can be expressed by
the number of variations in the sign of the Sturm chain at the bounds. Through
constructing the "Sturm matrix", a symmetric matrix associated with $f(x)$ over
$\mathbb R[x]$, variations in the sign of $f(x)$ can be characterized by the
negative index of inertia. Therefore, this paper offers a new proof of Sturm's
theorem using matrix theory.
