Computing Large Matrix Powers by Diagonalization

Why Diagonalization Helps with Powers

Repeatedly multiplying a matrix by itself can make a large power tedious to calculate, even when the entries of the original matrix are small. Diagonalization offers a different route: in a suitable coordinate system, the matrix acts by scaling each coordinate separately. A power then raises those scaling factors to the desired exponent. The final change back to the original coordinates requires matrix multiplication, but it avoids carrying out a long sequence of repeated products.

Throughout, matrices and scalars are real. Let \(\mathbf{A}\in\mathbb{R}^{n\times n}\). A diagonalization of \(\mathbf{A}\) is a factorization \(\mathbf{A}=\mathbf{P}\mathbf{D}\mathbf{P}^{-1}\), where \(\mathbf{P}\) is invertible and \(\mathbf{D}\) is diagonal. In the eigenvector interpretation, the columns of \(\mathbf{P}\), in order, are eigenvectors of \(\mathbf{A}\), and the corresponding eigenvalues appear in the same order on the diagonal of \(\mathbf{D}\). The order must match: changing the order of eigenvectors without making the corresponding change to \(\mathbf{D}\) will generally break the factorization.

For \(k=0\), the convention is \(\mathbf{A}^0=\mathbf{I}_n\), and the result is \(\mathbf{I}_n\) without any calculation. For positive \(k\), each diagonal entry is raised to \(k\), including a zero diagonal entry, whose positive integer power is zero. This distinction avoids treating the scalar expression \(0^0\) as an ordinary arithmetic operation.

Why the Power Formula Works

The matrix identity can be justified directly from the definition of a diagonalization. For positive powers, every adjacent pair \(\mathbf{P}^{-1}\mathbf{P}\) in the repeated product equals \(\mathbf{I}_n\). The following proof makes this cancellation precise, including the first positive power.

Proof. We use induction on \(k\). When \(k=1\), the assertion is the given equality \(\mathbf{A}=\mathbf{P}\mathbf{D}\mathbf{P}^{-1}\). Suppose it holds for a positive integer \(k\), so \(\mathbf{A}^k=\mathbf{P}\mathbf{D}^k\mathbf{P}^{-1}\). Then, using the definition of a matrix power and \(\mathbf{P}^{-1}\mathbf{P}=\mathbf{I}_n\),

Thus the formula holds for \(k+1\). By induction, it holds for every positive integer \(k\). \(\square\)

There is a useful entrywise version of the same calculation. Write \(p_{i\ell}\) for the entry in row \(i\), column \(\ell\) of \(\mathbf{P}\), and \(q_{\ell j}\) for the entry in row \(\ell\), column \(j\) of \(\mathbf{P}^{-1}\). For positive \(k\), the \((i,j)\) entry of the power is a sum of contributions from the diagonal coordinates:

Proof. By the theorem, \(\mathbf{A}^k=\mathbf{P}\mathbf{D}^k\mathbf{P}^{-1}\). The \((i,j)\) entry of a product of three matrices is obtained by summing over the row and column indices used to multiply them. Since \(\mathbf{D}^k\) is diagonal, its entry in row \(\ell\), column \(r\) is zero unless \(\ell=r\); when \(\ell=r\), that entry is \(\lambda_\ell^k\). Therefore only the terms with matching indices remain, giving \((\mathbf{A}^k)_{ij}=\sum_{\ell=1}^n p_{i\ell}\lambda_\ell^k q_{\ell j}\). \(\square\)

A Reliable Calculation Procedure

Worked Calculations

Common Error: Leaving Out the Inverse

The reconstruction is \(\mathbf{P}\mathbf{D}^k\mathbf{P}^{-1}\), not \(\mathbf{P}\mathbf{D}^k\mathbf{P}\). For the first example, use \(k=2\). Since \(\mathbf{D}^2=\begin{bmatrix}9&0\\0&1\end{bmatrix}\), incorrectly multiplying by \(\mathbf{P}\) on the right gives

But \(\mathbf{A}^2=\begin{bmatrix}2&1\\1&2\end{bmatrix}\begin{bmatrix}2&1\\1&2\end{bmatrix}=\begin{bmatrix}5&4\\4&5\end{bmatrix}\), so that calculation is wrong. The correction is to use \(\mathbf{P}^{-1}=\tfrac12\mathbf{P}\) on the right:

This example also shows why \(\mathbf{P}\) and \(\mathbf{P}^{-1}\) cannot be interchanged or omitted merely because they are related. In general, the inverse is essential to return from diagonal coordinates to the original coordinates.

When the Method Applies

The calculation applies when the matrix has a diagonalization over the chosen scalar field. If no such factorization is available, this procedure does not provide a diagonal form for computing the power; one must not assume that every matrix is diagonalizable. When a diagonalization is given, the diagonal entries can be powered independently, even if some eigenvalues repeat or are zero. For \(k\ge1\), a zero diagonal entry remains zero; for \(k=0\), the matrix power is the identity by definition.

For a large exponent, the benefit is most pronounced when the eigenvalues have simple powers or when many different powers of the same matrix are needed. The matrix \(\mathbf{P}\) and its inverse stay fixed while only the entries of \(\mathbf{D}^k\) change. The final multiplication may still involve arithmetic, but the repeated multiplication of the original matrix has been replaced by scalar powers and a single reconstruction.