Diagonalization

Diagonalization transforms a matrix into the simplest possible form—a diagonal matrix—by choosing an eigenvector basis. This process reveals the matrix's fundamental structure and enables efficient computation.

DefinitionDiagonalizable Matrix

An $n \times n$ matrix $A$ is diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that: $A = PDP^{-1} \quad \text{or equivalently} \quad P^{-1}AP = D$

The columns of $P$ are eigenvectors of $A$ , and the diagonal entries of $D$ are the corresponding eigenvalues.

When $A = PDP^{-1}$ , computing powers becomes trivial: $A^k = PD^kP^{-1}$ , and $D^k$ is just the diagonal entries raised to power $k$ .

TheoremDiagonalization Criterion

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

Equivalently, $A$ is diagonalizable if and only if the sum of geometric multiplicities equals $n$ : $\sum_{\lambda} \dim(E_\lambda) = n$

where the sum is over all distinct eigenvalues.

ExampleDiagonalizing a Matrix

Diagonalize $A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$ .

Step 1: Find eigenvalues from $\det(A - \lambda I) = 0$ : $(4-\lambda)(3-\lambda) - 2 = \lambda^2 - 7\lambda + 10 = 0$ Eigenvalues: $\lambda_1 = 5$ , $\lambda_2 = 2$ .

Step 2: Find eigenvectors. For $\lambda_1 = 5$ : $\mathbf{v}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$ For $\lambda_2 = 2$ : $\mathbf{v}_2 = \begin{pmatrix} 1 \\ -2 \end{pmatrix}$

Step 3: Form $P = [\mathbf{v}_1 \mid \mathbf{v}_2] = \begin{bmatrix} 1 & 1 \\ 1 & -2 \end{bmatrix}$ and $D = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix}$ .

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

TheoremSufficient Conditions for Diagonalizability

A matrix $A$ is guaranteed to be diagonalizable if:

$A$ has $n$ distinct eigenvalues, or
$A$ is symmetric (or Hermitian for complex matrices), or
For each eigenvalue $\lambda$ , the geometric multiplicity equals the algebraic multiplicity

DefinitionSimilar Matrices

Matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $B = P^{-1}AP$ .

Similar matrices represent the same linear transformation in different bases. They share:

Eigenvalues (with same multiplicities)
Characteristic polynomial
Determinant
Trace
Rank

Remark

Not all matrices are diagonalizable. For example, $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ has only one eigenvalue ( $\lambda = 1$ ) with geometric multiplicity 1, but algebraic multiplicity 2. Such matrices are handled by Jordan canonical form, which generalizes diagonalization.