A
[matrix polynomial](http://en.wikipedia.org/wiki/Matrix_polynomial)
is a polynomial whose variables are square $n \times n$ matrices,
let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$.
I am seeking a source of results on solving such equations.

For example, $X^2 =0$ has infinitely many solutions, because, e.g.,
$$ X =
\left( \begin{array}{ccc}
0 & 0 & x \\
0 & 0 & 0 \\
0 & 0 & 0 \end{array} \right)
$$
is a solution for all $x \in \mathbb{C}$.

Whereas
$$ X^2 -
\left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 9 & 0 \\
0 & 0 & 13 \end{array} \right)
= 0
$$
has $8$ solutions (and I believe no others),
$$ X =
\left( \begin{array}{ccc}
\pm 1 & 0 & 0 \\
0 & \pm 3 & 0 \\
0 & 0 & \pm \sqrt{13} \end{array} \right) \;.
$$

There is such a rich set of results on roots of polynomial equations over $\mathbb{C}$
and over $\mathbb{R}$, I am hoping there are analogs when the variables are matrices
rather than single elements of fields. But my (superficial) explorations did
not uncover a comprehensive source on this topic.

To pose a specific question:

> **Q**. Are there theorems that yield the number of solutions/roots
for such matrix polynomial equations?