Matrix Powers via
Diagonalization

Learn how to compute $A^n$ efficiently by decomposing a matrix into its eigenvalues and eigenvectors: the key technique that turns repeated multiplication into simple exponentiation.

Linear Algebra

What Is Diagonalization?

Diagonalization is the process of finding an invertible matrix $P$ and a diagonal matrix $D$ such that a square matrix $A$ can be written as:

$$A = PDP^{-1}$$

$P$

Matrix whose columns are the eigenvectors of $A$

$D$

Diagonal matrix with eigenvalues $\lambda_1, \lambda_2, \dots$ on the diagonal

$P^{-1}$

The inverse of the eigenvector matrix $P$

Why Compute Matrix Powers?

Computing $A^n$ arises in many real-world applications. Multiplying $A$ by itself $n$ times is slow: it costs $O(n \cdot d^3)$ operations. Diagonalization reduces this to a single matrix multiply plus raising scalars to powers.

Dynamical Systems

State after $n$ time steps: $\mathbf{x}_n = A^n \mathbf{x}_0$

Markov Chains

Long-run probabilities from transition matrix powers

Fibonacci Numbers

Closed-form via $2 \times 2$ matrix exponentiation

The Key Insight: Powers Simplify

When $A = PDP^{-1}$, multiplying $A$ by itself causes the inner terms to cancel. Watch what happens with $A^2$:

$$A^2 = (PDP^{-1})(PDP^{-1}) = PD\underbrace{(P^{-1}P)}_{I}DP^{-1} = PD^2P^{-1}$$

The Diagonalization Theorem

Diagonalizable vs. Not Diagonalizable

	Diagonalizable	Not Diagonalizable
Eigenvectors	$n$ linearly independent	Fewer than $n$ independent
Eigenvalues	Distinct (sufficient)	Repeated with deficient eigenspaces
Example	$\begin{pmatrix}2&0\\0&3\end{pmatrix}$	$\begin{pmatrix}2&1\\0&2\end{pmatrix}$
Computing $A^n$	$PD^nP^{-1}$	Use Jordan form or other methods

How to Compute $A^n$: Step by Step

1

Find eigenvalues. Solve $\det(A - \lambda I) = 0$ to get $\lambda_1, \lambda_2, \dots$

2

Find eigenvectors. For each $\lambda_i$, solve $(A - \lambda_i I)\mathbf{v} = \mathbf{0}$

3

Build $P$ and $D$. Columns of $P$ are eigenvectors; diagonal of $D$ holds eigenvalues (in matching order)

4

Compute $P^{-1}$. Use the inverse formula for the size of $P$

5

Assemble: $A^n = P \cdot \text{diag}(\lambda_1^n, \lambda_2^n, \dots) \cdot P^{-1}$

Example 1: Find $A^n$ for a $2 \times 2$ Matrix

Let $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$. We want to find a general formula for $A^n$.

Example 1: Assembling the Result

Example 2: Fibonacci Matrix

The Fibonacci recurrence $F_{n+1} = F_n + F_{n-1}$ can be written as a matrix equation:

$$\begin{pmatrix}F_{n+1}\\F_n\end{pmatrix} = \begin{pmatrix}1&1\\1&0\end{pmatrix}^n \begin{pmatrix}1\\0\end{pmatrix}$$

Example 3: A Specific Power

Compute $A^4$ for $A = \begin{pmatrix}5&4\\1&2\end{pmatrix}$.

Visualizing the Decomposition

Properties of $D^n$

Since $D$ is diagonal, its powers are trivially computed. Each entry is independent:

$$D^n = \begin{pmatrix}\lambda_1^n & 0 & \cdots & 0 \\ 0 & \lambda_2^n & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_k^n\end{pmatrix}$$

$|\lambda| > 1$

$\lambda^n$ grows: the component along that eigenvector expands

$|\lambda| = 1$

$\lambda^n$ stays bounded: the component oscillates or stays fixed

$|\lambda| < 1$

$\lambda^n \to 0$: the component decays to zero

When Can You Diagonalize?

✓

Distinct eigenvalues: If all $n$ eigenvalues are different, $A$ is always diagonalizable

✓

Symmetric/Hermitian matrices: Real symmetric matrices are always diagonalizable (via orthogonal eigenvectors)

?

Repeated eigenvalues: Check if geometric multiplicity equals algebraic multiplicity for each eigenvalue

✕

Defective matrices: If any eigenvalue has fewer independent eigenvectors than its algebraic multiplicity, diagonalization fails

Common Mistakes to Avoid

✕ Wrong Column Order

The columns of $P$ must match the order of eigenvalues on the diagonal of $D$. Swapping columns of $P$ without swapping diagonal entries gives incorrect results.

✕ Forgetting $P^{-1}$

Writing $A^n = PD^n$ and omitting the $P^{-1}$ factor. The full formula is $A^n = PD^nP^{-1}$: all three pieces are essential.

✕ Assuming All Matrices Diagonalize

Not every matrix is diagonalizable. Always verify you have $n$ linearly independent eigenvectors before applying the method.

Summary: The Full Pipeline

Step	Operation	Result
1	$\det(A - \lambda I) = 0$	Eigenvalues $\lambda_1, \dots, \lambda_n$
2	$(A - \lambda_i I)\mathbf{v} = \mathbf{0}$	Eigenvectors $\mathbf{v}_1, \dots, \mathbf{v}_n$
3	Form $P$ and $D$	$P = [\mathbf{v}_1 \| \cdots \| \mathbf{v}_n]$, $D = \text{diag}(\lambda_i)$
4	Invert $P$	$P^{-1}$
5	Multiply	$A^n = PD^nP^{-1}$

Question 1 of 5

If $A = PDP^{-1}$, which expression correctly gives $A^3$?

Since $A^n = PD^nP^{-1}$, for $n = 3$ we get $A^3 = PD^3P^{-1}$. The matrix $P$ appears once on the left and $P^{-1}$ once on the right: the intermediate $P^{-1}P$ pairs cancel when expanding $A \cdot A \cdot A$.

Question 2 of 5

A $3 \times 3$ matrix has eigenvalues $\lambda_1 = 2$, $\lambda_2 = -1$, $\lambda_3 = 4$. What is $D^5$?

$D^5 = \text{diag}(\lambda_1^5, \lambda_2^5, \lambda_3^5) = \text{diag}(2^5, (-1)^5, 4^5) = \text{diag}(32, -1, 1024)$. Each diagonal entry is raised to the power independently. Note that $(-1)^5 = -1$ because 5 is odd.

Question 3 of 5

Which matrix is not diagonalizable?

The matrix $\begin{pmatrix}5&1\\0&5\end{pmatrix}$ has a repeated eigenvalue $\lambda = 5$ with algebraic multiplicity 2, but its eigenspace is only 1-dimensional (spanned by $\begin{pmatrix}1\\0\end{pmatrix}$). Since $\text{geometric multiplicity} < \text{algebraic multiplicity}$, the matrix is defective and cannot be diagonalized.

Question 4 of 5

For $A = \begin{pmatrix}2&1\\0&3\end{pmatrix}$, what is $A^3$?

From Example 1, we derived $A^n = \begin{pmatrix}2^n & 3^n - 2^n\\0 & 3^n\end{pmatrix}$. Substituting $n = 3$: the (1,1) entry is $2^3 = 8$, the (1,2) entry is $3^3 - 2^3 = 27 - 8 = 19$, the (2,2) entry is $3^3 = 27$. So $A^3 = \begin{pmatrix}8&19\\0&27\end{pmatrix}$.

Question 5 of 5

A diagonalizable matrix $A$ has eigenvalues $\lambda_1 = 0.5$ and $\lambda_2 = 0.8$. What happens to $A^n$ as $n \to \infty$?

Since both $|\lambda_1| = 0.5 < 1$ and $|\lambda_2| = 0.8 < 1$, we have $\lambda_1^n \to 0$ and $\lambda_2^n \to 0$ as $n \to \infty$. Therefore $D^n \to \mathbf{0}$, and $A^n = PD^nP^{-1} \to P \cdot \mathbf{0} \cdot P^{-1} = \mathbf{0}$. All eigenvalues with absolute value less than 1 decay to zero.

Step	Operation	Result
1	\(\det(A - \lambda I) = 0\)	Eigenvalues \(\lambda_1, \dots, \lambda_n\)
2	\((A - \lambda_i I)\mathbf{v} = \mathbf{0}\)	Eigenvectors \(\mathbf{v}_1, \dots, \mathbf{v}_n\)
3	Form \(P\) and \(D\)	\(P = [\mathbf{v}_1 \| \cdots \| \mathbf{v}_n]\), \(D = \text{diag}(\lambda_i)\)
4	Invert \(P\)	\(P^{-1}\)
5	Multiply	\(A^n = PD^nP^{-1}\)

Matrix Powers via
Diagonalization

What Is Diagonalization?

Why Compute Matrix Powers?

The Key Insight: Powers Simplify

The Diagonalization Theorem

Diagonalizable vs. Not Diagonalizable

How to Compute \(A^n\): Step by Step

Example 1: Find \(A^n\) for a \(2 \times 2\) Matrix

Example 1: Assembling the Result

Example 2: Fibonacci Matrix

Example 3: A Specific Power

Visualizing the Decomposition

Properties of \(D^n\)

When Can You Diagonalize?

Common Mistakes to Avoid

Summary: The Full Pipeline

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

	Diagonalizable	Not Diagonalizable
Eigenvectors	\(n\) linearly independent	Fewer than \(n\) independent
Eigenvalues	Distinct (sufficient)	Repeated with deficient eigenspaces
Example	\(\begin{pmatrix}2&0\\0&3\end{pmatrix}\)	\(\begin{pmatrix}2&1\\0&2\end{pmatrix}\)
Computing \(A^n\)	\(PD^nP^{-1}\)	Use Jordan form or other methods

Matrix Powers viaDiagonalization

What Is Diagonalization?

Why Compute Matrix Powers?

The Key Insight: Powers Simplify

The Diagonalization Theorem

Diagonalizable vs. Not Diagonalizable

How to Compute \(A^n\): Step by Step

Example 1: Find \(A^n\) for a \(2 \times 2\) Matrix

Example 1: Assembling the Result

Example 2: Fibonacci Matrix

Example 3: A Specific Power

Visualizing the Decomposition

Properties of \(D^n\)

When Can You Diagonalize?

Common Mistakes to Avoid

Summary: The Full Pipeline

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Matrix Powers via
Diagonalization