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Diagonalization

A matrix $A$ is diagonalizable if it can be written as $A = PDP^{-1}$ , where $D$ is a diagonal matrix and $P$ is invertible. The columns of $P$ are eigenvectors and the diagonal entries of $D$ are the corresponding eigenvalues. This decomposition transforms $A$ into the simplest possible form — just scaling along eigenvector directions.

Diagonalization makes computing powers trivially easy. Since $A^k = PD^kP^{-1}$ , and $D^k$ just raises each diagonal entry to the $k$ -th power: $D^k = \text{diag}(\lambda_1^k, \ldots, \lambda_n^k)$ . For a $1000 \times 1000$ matrix applied 50 times, diagonalization replaces $10^9$ multiplications with $1000$ scalar exponentiations.

An $n \times n$ matrix is diagonalizable if and only if it has $n$ linearly independent eigenvectors — equivalently, if the geometric multiplicity equals the algebraic multiplicity for every eigenvalue. Use the interactive diagram to explore: try the "Distinct λ" and "Symmetric" presets (both diagonalizable) versus "Not diag." (defective) and "Rotation" (complex eigenvalues).

Formal View

Theorem 7.12 — Diagonalization Theorem

n \times n

matrix

A

is diagonalizable if and only if it has

n

linearly independent eigenvectors

\mathbf{v}_1, \ldots, \mathbf{v}_n

. When diagonalizable:

A = PDP^{-1}

where

P = [\mathbf{v}_1 \mid \cdots \mid \mathbf{v}_n]

and

D = \text{diag}(\lambda_1, \ldots, \lambda_n)

Corollary 7.13

A

has

n

distinct eigenvalues, then

A

is diagonalizable. (The converse is false: a matrix can be diagonalizable even with repeated eigenvalues.)

Remark 7.14

Powers formula:

A^k = PD^kP^{-1}

for any positive integer

k

. More generally, for any polynomial

f

, we have

f(A) = Pf(D)P^{-1}

where

f(D) = \text{diag}(f(\lambda_1), \ldots, f(\lambda_n))

Interactive Visualization

Diagonalization: A = PDP⁻¹

Why This Matters

Diagonalization is the key to computing matrix powers, solving linear ODEs, and understanding long-term behavior of dynamical systems.

Markov chains: $A^k$ converges to a steady-state as $k \to \infty$ when the dominant eigenvalue is 1 and all others are strictly inside the unit circle
Image and signal processing: diagonal operations in the eigenbasis enable fast filtering and compression
Quantum computing: quantum gates are often described by their spectral decomposition
Population dynamics: long-run growth rates of populations are determined by the dominant eigenvalue of the Leslie matrix

Learning Resources

Diagonalization and powers of a matrix

MIT OpenCourseWare

Gilbert Strang explains the diagonalization A = PDP⁻¹ and its application to matrix powers.

51 min

Diagonalizing a matrix

Khan Academy

Step-by-step worked example of finding P, D, and verifying PDP⁻¹ = A.

18 min

Quiz

Question 1

If $A = PDP^{-1}$ , what is $A^3$ ?

Question 2

A matrix with a repeated eigenvalue is never diagonalizable.

Question 3

In the diagonalization $A = PDP^{-1}$ , the columns of $P$ are:

Common Mistakes

Writing $A^k = P^k D^k P^{-k}$ — the formula is $PD^kP^{-1}$ , not $P^kD^kP^{-k}$ .
Confusing diagonalizable with invertible — a matrix can be diagonalizable with eigenvalue 0 (not invertible), or invertible but not diagonalizable (defective).
Forgetting to check that $P$ is invertible — if the eigenvectors are linearly dependent, the diagonalization fails.