Recipe

Compute $A^k$ by diagonalization

Use when: you have (or can find) $A=PDP^{-1}$ and need $A^k$ for some integer $k$ , rather than multiplying $A$ by itself $k$ times

Goal: $A^k$ , computed as $PD^kP^{-1}$ at a cost that doesn't grow with $k$

Taught in Powers of a matrix by diagonalization · extends Diagonalize a matrix

Quick recipe

Diagonalize $A$ as $PDP^{-1}$
Raise $D$ to the $k$ -th power, entry by entry
Assemble $A^k = PD^kP^{-1}$
Sanity-check the result against the trace

1
If you don't already have $P$ , $D$ , and $P^{-1}$ , find them exactly as in the diagonalize-a-matrix recipe: the eigenvalues, a matching eigenvector for each, checked for independence, assembled into $P$ and $D$ in the same column order, then $P^{-1}$ .

Example

For $A = \begin{bmatrix} 4 & -1 \\ 2 & 1 \end{bmatrix}$ : the eigenvalues are $\lambda_1=3$ , $\lambda_2=2$ , giving $D=\begin{bmatrix}3&0\\0&2\end{bmatrix}$ , with matching eigenvectors assembled into $P=\begin{bmatrix}1&1\\1&2\end{bmatrix}$ and $P^{-1}=\begin{bmatrix}2&-1\\-1&1\end{bmatrix}$ .

Why: $A^k=PD^kP^{-1}$ only holds once $A=PDP^{-1}$ actually does, so this step's work is identical to the diagonalize-a-matrix recipe's own output.
2
Because $D$ is diagonal, $D^k$ just raises every diagonal entry to the $k$ -th power — no cross terms, no matrix multiplication needed.

Example

$D^4 = \begin{bmatrix} 3^4 & 0 \\ 0 & 2^4 \end{bmatrix} = \begin{bmatrix} 81 & 0 \\ 0 & 16 \end{bmatrix}$
- $k$ is a positive integer — raise each diagonal entry directly.
- $k$ is negative and every eigenvalue is nonzero — $A$ is invertible; reciprocate each entry (or raise it to the negative power) the same way.
Why: this is the entire reason diagonalization helps here — a diagonal matrix's $k$ -th power costs nothing more than exponentiating $n$ plain numbers, unlike repeated multiplication of the full matrix $A$ .
3
Multiply $P$ , $D^k$ , and $P^{-1}$ together in that order — it's most efficient to compute $D^kP^{-1}$ first, then left-multiply the result by $P$ .

Example

$D^4P^{-1} = \begin{bmatrix} 81 & 0 \\ 0 & 16 \end{bmatrix}\begin{bmatrix} 2 & -1 \\ -1 & 1 \end{bmatrix} = \begin{bmatrix} 162 & -81 \\ -16 & 16 \end{bmatrix}, \qquad A^4 = \begin{bmatrix} 1 & 1 \\ 1 & 2 \end{bmatrix}\begin{bmatrix} 162 & -81 \\ -16 & 16 \end{bmatrix} = \begin{bmatrix} 146 & -65 \\ 130 & -49 \end{bmatrix}$

Why: this is exactly the identity $A^k=(PDP^{-1})^k=PD^kP^{-1}$ , every interior $P^{-1}P$ pair having already telescoped away.
4
Add the diagonal entries of the computed $A^k$ and confirm the sum equals $\lambda_1^k+\cdots+\lambda_n^k$ .

Example

$\operatorname{tr}(A^4) = 146+(-49)=97$ , and $\lambda_1^4+\lambda_2^4 = 81+16=97$ — the two match.

Why: the trace of a power always equals the sum of the eigenvalues raised to that power, catching an arithmetic slip in the matrix multiplication without redoing the whole computation.

Common mistakes

Raising $A$ itself to the $k$ -th power somewhere in the arithmetic instead of raising only $D$ 's diagonal entries.
Multiplying in the wrong order, e.g. $P^{-1}D^kP$ , which doesn't telescope back to $A^k$ .
Trying a negative power when one eigenvalue is $0$ — a matrix with a zero eigenvalue isn't invertible, so no negative power of it exists.
Skipping diagonalization entirely and assuming this shortcut applies to any matrix — it requires $A$ to actually be diagonalizable first.

Compute A^k by diagonalization

Compute $A^k$ by diagonalization