A 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?

A 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 $x \in \mathbb{C}$.

(Second example removed, as it did not fit the definition.)

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?

A 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 yields the number of solutions/roots for such matrix polynomial equations?

A 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?