Matrix Polynomials

Suppose that we have a polynomial in the form $f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$ where $a_0, a_1, ..., a_n$ are constants. If $A$ is an $n \times n$ square matrix, then we can define $f(A) = a_0I_n + a_1A + a_2A^2 + ... + a_nA^n$.

For example, consider the polynomial function $f(x) = 2 + 3x + x^2$, and let $A = \begin{bmatrix} 3 & 4\\ 1 & 1 \end{bmatrix}$. We will substitute $x = A$ and thus compute $f(A)$ using matrix arithmetic:

(1)

\begin{align} f(A) = 2\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} + 3\begin{bmatrix} 3 & 4\\ 1 & 1 \end{bmatrix} + \left ( \begin{bmatrix} 3 & 4\\ 1 & 1 \end{bmatrix} \right )^2 \\ f(A) = \begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix} + \begin{bmatrix} 9 & 12\\ 3 & 3 \end{bmatrix} + \begin{bmatrix} 13 & 16\\ 4 & 5 \end{bmatrix} \\ f(A) = \begin{bmatrix} 24 & 28\\ 7 & 10 \end{bmatrix} \end{align}

Example 1

Given the polynomial function $p(x) = 2x - x^2 + 3$, compute $p(B)$ given that $B = \begin{bmatrix} 7 & -2\\ 0 & 2 \end{bmatrix}$.

(2)

\begin{align} p(B) = 2\begin{bmatrix} 7 & -2\\ 0 & 2 \end{bmatrix} - \left ( \begin{bmatrix} 7 & -2\\ 0 & 2 \end{bmatrix} \right )^2 + 3\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \\ p(B) = \begin{bmatrix} 14 & -4\\ 0 & 4 \end{bmatrix} - \begin{bmatrix} 49 & -18\\ 0 & 4 \end{bmatrix} + \begin{bmatrix} 3 & 0\\ 0 & 3 \end{bmatrix} \\ p(B) = \begin{bmatrix} 66 & -22 \\ 0 & 11 \end{bmatrix} \end{align}