Diagonalization

You have learned that eigenvalues and eigenvectors reveal the special directions where a matrix acts simply, just by scaling. Now comes the payoff: if a matrix has enough eigenvectors, you can completely simplify how you work with it. Instead of dealing with a complicated transformation, you can change your perspective to one where the matrix becomes diagonal, with all its action happening along the coordinate axes. This process is called diagonalization, and it transforms difficult matrix computations into trivially easy ones.

Think about driving through a city on a grid of streets versus driving through a maze of winding roads. The grid makes navigation simple because everything aligns with your natural sense of “forward” and “sideways.” Diagonalization does something similar: it finds a coordinate system where the matrix’s action is as simple as possible, stretching or compressing along perpendicular directions with no mixing between them.

In this lesson, you will learn when a matrix can be diagonalized, how to find the change-of-basis matrix that accomplishes this, and why this is so powerful for computing matrix powers and solving applied problems.

Core Concepts

What Is Diagonalization?

A square matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that

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

Equivalently, you can rearrange this as $P^{-1}AP = D$, which shows that $D$ is what $A$ “looks like” when you change to a new coordinate system described by $P$.

The matrices in this equation have specific meanings:

• $D$ is a diagonal matrix with the eigenvalues of $A$ on its diagonal
• $P$ is formed by placing the corresponding eigenvectors of $A$ as its columns
• The columns of $P$ must be in the same order as the eigenvalues in $D$

For example, if $A$ has eigenvalues $\lambda_1 = 3$ and $\lambda_2 = 5$ with corresponding eigenvectors $\vec{v}_1$ and $\vec{v}_2$, then:

$$D = \begin{pmatrix} 3 & 0 \ 0 & 5 \end{pmatrix}, \quad P = \begin{pmatrix} | & | \ \vec{v}_1 & \vec{v}_2 \ | & | \end{pmatrix}$$

The equation $A = PDP^{-1}$ tells you that applying $A$ is equivalent to: first changing coordinates (multiply by $P^{-1}$), then scaling along the axes by the eigenvalues (multiply by $D$), and finally changing back to the original coordinates (multiply by $P$).

When Is a Matrix Diagonalizable?

Not every matrix can be diagonalized. The key requirement is having enough linearly independent eigenvectors.

Diagonalizability Criterion: An $n \times n$ matrix $A$ is diagonalizable if and only if $A$ has $n$ linearly independent eigenvectors.

This happens in several important cases:

1. Distinct eigenvalues: If $A$ has $n$ distinct eigenvalues, then $A$ is definitely diagonalizable. Eigenvectors corresponding to different eigenvalues are always linearly independent.

2. Repeated eigenvalues with full eigenspaces: If an eigenvalue $\lambda$ has algebraic multiplicity $k$ (it appears $k$ times as a root of the characteristic polynomial), then $A$ is diagonalizable only if the geometric multiplicity (dimension of the eigenspace) also equals $k$. You need $k$ linearly independent eigenvectors for that eigenvalue.

3. Symmetric matrices: Real symmetric matrices are always diagonalizable. Even better, they can be diagonalized using an orthonormal basis of eigenvectors.

When a matrix fails to be diagonalizable, it is called defective. This happens when there are not enough eigenvectors to form a basis. The classic example is the matrix $\begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}$, which has only one eigenvalue ($\lambda = 1$) with algebraic multiplicity 2, but only one linearly independent eigenvector.

Similar Matrices

Two matrices $A$ and $B$ are called similar if there exists an invertible matrix $P$ such that

$$B = P^{-1}AP$$

Similar matrices represent the same linear transformation, just described in different coordinate systems. They share many important properties:

• Same eigenvalues (with the same multiplicities)
• Same determinant
• Same trace
• Same rank

When we diagonalize $A$, we are finding a diagonal matrix $D$ that is similar to $A$. The diagonal matrix is the simplest representative of all matrices similar to $A$.

The Diagonalization Process

Here is how to diagonalize a matrix $A$:

Step 1: Find the eigenvalues.

Solve the characteristic equation $\det(A - \lambda I) = 0$.

Step 2: Find the eigenvectors.

For each eigenvalue $\lambda_i$, solve $(A - \lambda_i I)\vec{v} = \vec{0}$ to find the eigenvectors.

Step 3: Check if diagonalization is possible.

Verify that you have $n$ linearly independent eigenvectors total. If not, $A$ is not diagonalizable.

Step 4: Form $P$ and $D$.

• Place the eigenvectors as columns of $P$
• Place the corresponding eigenvalues on the diagonal of $D$ (in the same order)

Step 5: Verify (optional but recommended).

Check that $AP = PD$, which is equivalent to $A = PDP^{-1}$.

Powers of Diagonal Matrices

One of the main reasons diagonalization is so useful is that powers of diagonal matrices are trivially easy to compute. If

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

then

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

You simply raise each diagonal entry to the power $k$. There is no mixing between entries because the off-diagonal entries are all zero.

Computing $A^n$ Using Diagonalization

Here is where diagonalization really pays off. If $A = PDP^{-1}$, then:

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

The middle terms $P^{-1}P$ collapse to the identity matrix. Continuing this pattern:

$$A^n = PD^nP^{-1}$$

This is enormously powerful. Instead of multiplying $A$ by itself $n$ times (which would require $n-1$ matrix multiplications), you:

1. Compute $D^n$ by raising each diagonal entry to the $n$-th power
2. Multiply $PD^nP^{-1}$ (just two matrix multiplications, regardless of $n$)

For large $n$, this is vastly more efficient. Computing $A^{100}$ directly would require 99 matrix multiplications. Using diagonalization, it requires the same amount of work as computing $A^2$.

Symmetric Matrices and Orthogonal Diagonalization

Real symmetric matrices (where $A = A^T$) have two remarkable properties:

1. All eigenvalues are real. Even though the characteristic polynomial might seem like it could have complex roots, symmetry guarantees they are all real numbers.

2. Eigenvectors for different eigenvalues are orthogonal. If $\vec{v}_1$ and $\vec{v}_2$ are eigenvectors for different eigenvalues, then $\vec{v}_1 \cdot \vec{v}_2 = 0$.

This means symmetric matrices can always be diagonalized, and moreover, you can choose the eigenvectors to form an orthonormal basis. When you do this, the matrix $P$ becomes an orthogonal matrix, meaning $P^{-1} = P^T$.

This is called orthogonal diagonalization:

$$A = PDP^T$$

where $P$ is orthogonal (its columns are orthonormal eigenvectors) and $D$ is diagonal.

Orthogonal diagonalization is particularly nice because computing $P^{-1}$ is as easy as taking the transpose, and orthogonal matrices preserve lengths and angles.

Notation and Terminology

Term	Meaning	Example
Diagonalizable	Can be written as $PDP^{-1}$ where $D$ is diagonal	A matrix with $n$ linearly independent eigenvectors
Similar matrices	$B = P^{-1}AP$ for some invertible $P$	Similar matrices have the same eigenvalues
$P$	Matrix of eigenvectors (as columns)	Columns are eigenvectors in order matching $D$
$D$	Diagonal matrix of eigenvalues	Eigenvalues on diagonal, zeros elsewhere
Defective matrix	Not diagonalizable	Has fewer than $n$ independent eigenvectors
Orthogonal diagonalization	$A = PDP^T$ with orthogonal $P$	Possible for symmetric matrices

Examples

Key Properties and Rules

Summary of Diagonalization Properties

Property	Statement
Diagonalizability criterion	$A$ is diagonalizable if and only if it has $n$ linearly independent eigenvectors
Distinct eigenvalues	If all $n$ eigenvalues are distinct, then $A$ is diagonalizable
Powers formula	If $A = PDP^{-1}$, then $A^n = PD^nP^{-1}$
Similar matrices	$A$ and $D$ are similar: they have the same eigenvalues, determinant, and trace
Symmetric matrices	Real symmetric matrices are always diagonalizable
Orthogonal diagonalization	For symmetric $A$: $A = PDP^T$ where $P$ is orthogonal

The Diagonalization Algorithm

1. Compute eigenvalues: Solve $\det(A - \lambda I) = 0$
2. Compute eigenvectors: For each $\lambda$, solve $(A - \lambda I)\vec{v} = \vec{0}$
3. Check: Count total independent eigenvectors. Need exactly $n$ for an $n \times n$ matrix
4. Form matrices: $P$ = [eigenvectors as columns], $D$ = diagonal matrix of eigenvalues
5. Result: $A = PDP^{-1}$

When Diagonalization Fails

A matrix is NOT diagonalizable when:

• An eigenvalue has geometric multiplicity strictly less than its algebraic multiplicity
• Equivalently: the total count of linearly independent eigenvectors is less than $n$

Real-World Applications

Long-Term Behavior of Markov Chains

Markov chains model systems that transition between states with fixed probabilities, like weather patterns (sunny vs. rainy), customer behavior (loyal vs. churned), or web browsing (which page you visit next). The transition probabilities form a matrix, and the long-term distribution of states is determined by the eigenvalues.

If you want to know where the system will be after many steps, you need to compute a high power of the transition matrix. Diagonalization makes this tractable: instead of multiplying the matrix by itself hundreds of times, you raise the eigenvalues to that power. The eigenvalue equal to 1 (which always exists for valid Markov chains) determines the steady-state distribution, while eigenvalues less than 1 in magnitude represent transient effects that die out over time.

Solving Systems of Differential Equations

Many physical systems are described by systems of linear differential equations of the form $\vec{x}’(t) = A\vec{x}(t)$. When $A$ is diagonalizable, you can decouple these equations by changing to the eigenvector basis.

In the new coordinates, the system becomes a set of independent equations, each of the form $y’(t) = \lambda y(t)$. These have the simple solution $y(t) = y(0)e^{\lambda t}$. Transforming back gives the complete solution to the original system. This technique is fundamental in physics and engineering for analyzing oscillating systems, electrical circuits, and mechanical vibrations.

Population Dynamics (Leslie Matrices)

Ecologists use Leslie matrices to model how populations with different age groups evolve over time. The matrix entries represent birth rates and survival probabilities for each age class. To predict the population structure many generations in the future, you need to compute high powers of the Leslie matrix.

Diagonalization reveals the long-term growth rate (the largest eigenvalue) and the stable age distribution (the corresponding eigenvector). This helps ecologists understand whether a population will grow, shrink, or stabilize, and what the eventual age structure will look like.

Efficient Computation of Matrix Powers

Beyond the specific applications above, diagonalization is a general-purpose tool for computing matrix powers efficiently. Any problem that involves repeated application of a linear transformation, from image processing to economics to quantum mechanics, benefits from diagonalization.

The computational savings are dramatic. For an $n \times n$ matrix, computing $A^k$ directly requires about $k \cdot n^3$ operations. Using diagonalization, you need about $n^3$ operations to find $P$ and $P^{-1}$, then just $n$ operations to compute $D^k$ (raising each diagonal entry to the $k$-th power), and finally about $n^3$ operations to multiply $PD^kP^{-1}$. This is essentially independent of $k$, making it vastly more efficient for large powers.

Self-Test Problems

Summary

• A matrix $A$ is diagonalizable if it can be written as $A = PDP^{-1}$ where $D$ is a diagonal matrix of eigenvalues and $P$ is a matrix whose columns are the corresponding eigenvectors.

• When is a matrix diagonalizable? An $n \times n$ matrix is diagonalizable if and only if it has $n$ linearly independent eigenvectors. This is guaranteed when all eigenvalues are distinct. When eigenvalues repeat, you need the geometric multiplicity to equal the algebraic multiplicity.

• The diagonalization process:

  1. Find all eigenvalues by solving $\det(A - \lambda I) = 0$
  2. Find eigenvectors for each eigenvalue by solving $(A - \lambda I)\vec{v} = \vec{0}$
  3. Form $P$ (eigenvectors as columns) and $D$ (eigenvalues on diagonal, in matching order)

• Powers of diagonal matrices are easy: $D^n$ is found by raising each diagonal entry to the $n$-th power.

• Computing $A^n$: If $A = PDP^{-1}$, then $A^n = PD^nP^{-1}$. This is vastly more efficient than multiplying $A$ by itself $n$ times.

• Symmetric matrices are always diagonalizable. Their eigenvalues are real, and eigenvectors for different eigenvalues are orthogonal. They can be orthogonally diagonalized: $A = PDP^T$ where $P$ is orthogonal.

• Applications include computing long-term behavior of Markov chains, solving systems of differential equations, modeling population dynamics with Leslie matrices, and any situation requiring efficient computation of matrix powers.

• A matrix that is not diagonalizable is called defective. This happens when an eigenvalue does not have enough linearly independent eigenvectors.