Powers of a matrix (nonnegative exponents)

Section 3.6 Powers of a matrix (nonnegative exponents)

Subsection 3.6.1 Computing the powers of a square matrix \(A\)

If \(A\) and \(B\) are both square matrices of order \(n\text{,}\) then \(AB\) is also a square matrix of order \(n\text{.}\) In particular, if \(A=B\text{,}\) then \(AA\) is defined. We call this matrix \(A^2\text{.}\) We also let \(A^3=AAA\text{.}\) Similarly we have higher powers of \(A\text{:}\)

\begin{equation*} A^n=\underbrace{A A A \cdots A A}_{n \text{ factors}}\quad \text{ for } n=1,2,\ldots\text{.} \end{equation*}

In addition, we define \(A^1=A\text{.}\)

If \(A=\begin{bmatrix}1\amp2\\2\amp1\end{bmatrix}\) then it is straightforward to compute the powers of \(A\text{:}\)

Table 3.6.1. Powers of \(A=\bigl[\begin{smallmatrix}1\amp2\\2\amp1\end{smallmatrix}\bigr]\)

\(A^1=\begin{bmatrix}1\amp2\\2\amp1\end{bmatrix}\)

\(A^2=\begin{bmatrix}5\amp4\\4\amp5\end{bmatrix}\)

\(A^3=\begin{bmatrix}13\amp14\\14\amp13\end{bmatrix}\)

\(A^4=\begin{bmatrix}40\amp41\\41\amp40\end{bmatrix}\)

\(A^5=\begin{bmatrix}121\amp122\\122\amp121\end{bmatrix}\)

It is not easy to write a general expression for \(A^n\text{.}\) When we have developed more sophisticated tools, we will be able to do so.

Subsection 3.6.2 The law of exponents

If \(m\) and \(n\) are positive integers, then by simply counting the factors we get the following two equations:

\(\displaystyle A^m A^n=\underbrace{A\cdots A}_{m \text{ factors}}\ \underbrace{A\cdots A}_{n \text{ factors}}= \underbrace{A\cdots A}_{m+n\text{ factors}}=A^{m+n},\)
\(\displaystyle (A^m)^n =\underbrace{(\underbrace{A\cdots A}_{m \text{ factors}}) (\underbrace{A\cdots A}_{m \text{ factors}})\cdots (\underbrace{A\cdots A}_{m \text{ factors}})}_{n \text{ times}}=A^{mn}\)

These two equations together are called the law of exponents.

Subsection 3.6.3 What is \(A^0\text{?}\)

We want to define \(A^0\) so the the law of exponents remains valid. This says

\begin{equation*} A^n A^0= A^{n+0}=A^n \end{equation*}

We observe that if \(A^0=I\text{,}\) the identity matrix, then this equation is valid. With this in mind we \(\textbf{define}\) \(A^0=I\text{.}\)

Subsection 3.6.4 Polynomials and powers of a matrix

Recall that polynomials are functions of the form \(p(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_1x+a_0\text{.}\) If we have a square matrix, we may refer to \(p(A)\text{.}\) By this we mean we substitute the matrix \(A\) for each \(x\) appearing in the polynomial. Whenever \(x^k\) appears, we substitute \(A^k\) for it and do the computations. For example, using the same \(A\) as before, if \(p(x)=x^2-2x+1\text{,}\) then

\begin{equation*} p(A)=A^2-2A+I= \begin{bmatrix}5\amp4\\4\amp5\end{bmatrix} -2\begin{bmatrix}1\amp2\\2\amp1\end{bmatrix} +\begin{bmatrix}1\amp0\\0\amp1\end{bmatrix} = \begin{bmatrix}4\amp0\\0\amp4\end{bmatrix} \end{equation*}

Now consider the matrix

\begin{equation*} B=\begin{bmatrix}2\amp-1\\1\amp0\end{bmatrix} \end{equation*}

and the same polynomial \((x)=x^2-2x+1\text{.}\) In this case

\begin{equation*} p(B)=B^2-2B+I= \begin{bmatrix}3\amp-2\\2\amp-1\end{bmatrix} -2\begin{bmatrix}2\amp-1\\1\amp0\end{bmatrix} +\begin{bmatrix}1\amp0\\0\amp1\end{bmatrix} = \begin{bmatrix}0\amp0\\0\amp0\end{bmatrix} \end{equation*}

and so we get the (matrix) equation \(P(B)=0\text{.}\) When a matrix \(B\) satisfies the equation \(P(B)=0\text{,}\) we call \(B\) a root of \(p(x)\text{.}\)