Defining And Computing Matrix Powers

What a Matrix Power Means

Multiplying a square matrix by itself repeatedly gives a compact way to describe repeated applications of the same matrix. The notation \(\mathbf{A}^k\) records how many copies of \(\mathbf{A}\) are multiplied, while the case \(k=0\) is defined using the identity matrix. These conventions make powers useful for organising calculations, but the definition is simple: for a positive exponent, multiply the matrix by itself the specified number of times.

Throughout this tutorial, \(\mathbf{A}\) is an \(n\times n\) matrix with real entries, where \(n\geq 1\). The exponent \(k\) is a nonnegative integer, meaning \(k\in\{0,1,2,\ldots\}\). The square-matrix condition matters: only then can \(\mathbf{A}\) be multiplied by itself repeatedly and still have compatible dimensions. The identity matrix \(\mathbf{I}_n\) is the \(n\times n\) matrix that leaves every \(n\)-component vector unchanged when multiplied by it.

The recursive definition specifies a multiplication to perform at every step: start at \(\mathbf{A}^0=\mathbf{I}_n\), then multiply the current matrix on the right by \(\mathbf{A}\). In particular, each resulting power is again \(n\times n\), so the next multiplication is defined. The convention \(\mathbf{A}^0=\mathbf{I}_n\) is not an assertion that no matrix exists at exponent zero; it gives a specific matrix, the identity, as the zeroth power.

Why Repeated Multiplication Gives the Power

Proof. At \(k=0\), \(\mathbf{A}^0=\mathbf{I}_n\), which is \(n\times n\). Suppose \(\mathbf{A}^{k-1}\) is \(n\times n\), for some \(k\geq1\). Then the product \(\mathbf{A}^{k-1}\mathbf{A}\) is defined, since the number of columns of the first matrix and the number of rows of the second are both \(n\). Its dimensions are \(n\times n\), so \(\mathbf{A}^k\) is also \(n\times n\). This proves the dimension claim by induction. For the product description, the case \(k=1\) gives \(\mathbf{A}^1=\mathbf{A}^0\mathbf{A}=\mathbf{I}_n\mathbf{A}=\mathbf{A}\). If \(\mathbf{A}^{k-1}\) consists of \(k-1\) copies of \(\mathbf{A}\) grouped according to the recursive definition, then \(\mathbf{A}^k=\mathbf{A}^{k-1}\mathbf{A}\) appends one more copy. It therefore consists of \(k\) copies, as claimed. \(\square\)

The calculation can also be described entry by entry. If \(c_{ij}\) is the entry in row \(i\), column \(j\) of \(\mathbf{A}^k\), then it is obtained from the entries of \(\mathbf{A}^{k-1}\) and \(\mathbf{A}\) by the ordinary matrix multiplication rule.

Justification. By the definition of matrix multiplication, the entry in row \(i\), column \(j\) of a product \(\mathbf{B}\mathbf{C}\) is \(\sum_{r=1}^{n}b_{ir}c_{rj}\), when both matrices are \(n\times n\). Substitute \(\mathbf{B}=\mathbf{A}^{k-1}\) and \(\mathbf{C}=\mathbf{A}\). The resulting entry is precisely the displayed sum, so applying the usual matrix multiplication rule at each step computes the recursively defined power. \(\square\)

Computing Powers One Step at a Time

Do not replace matrix multiplication with multiplication of matching entries. For a product, each entry combines an entire row and column through a sum of products. The worked examples show how this affects powers, including the identity at exponent zero.

A Power Can Have Many Zero Entries

Common Mistakes and Their Corrections

Confusing the identity with the zero matrix. A frequent error is to set \(\mathbf{A}^0=\mathbf{O}\), the zero matrix. The definition instead gives \(\mathbf{A}^0=\mathbf{I}_n\). For example, with \(\mathbf{A}=\begin{bmatrix}1&2\\0&1\end{bmatrix}\), the correct value is \(\mathbf{A}^0=\begin{bmatrix}1&0\\0&1\end{bmatrix}\), not \(\begin{bmatrix}0&0\\0&0\end{bmatrix}\). The identity is the appropriate starting matrix because it has the same size as \(\mathbf{A}\) and leaves \(\mathbf{A}\) unchanged when multiplied by it.

Squaring entries instead of multiplying matrices. For \(\mathbf{D}=\begin{bmatrix}1&2\\3&4\end{bmatrix}\), squaring each entry would give \(\begin{bmatrix}1&4\\9&16\end{bmatrix}\). That is not \(\mathbf{D}^2\). The correct calculation is

Each entry comes from a row-by-column product, not from a single entry of \(\mathbf{D}\). As a check, the first entry is \(1+6=7\), and the last is \(6+16=22\); both differ from the corresponding entrywise squares.

Trying to form powers of a nonsquare matrix. Under this definition, repeated self-multiplication requires a square matrix. For instance, if \(\mathbf{E}\) is \(2\times3\), the product \(\mathbf{E}\mathbf{E}\) is not defined: the first factor has three columns, while the second has two rows. Check dimensions before starting a power calculation.

Keeping the Definition in View

For low powers, the most reliable approach is to write each multiplication explicitly and keep the previous result available. The zeroth power is the identity of the correct size; the first power is the matrix itself; each subsequent power is the preceding result multiplied by the original matrix. This definition gives a direct computational procedure without requiring shortcuts. When a calculation is complete, check that every product was dimensionally valid and that its entries came from row-by-column multiplication.