93. Diagonalization

A square matrix is diagonalizable if it can be written as

where is a diagonal matrix and is an invertible matrix.

The diagonal entries of are the eigenvalues of , and the columns of are the corresponding eigenvectors.

93.1. When is diagonalizable?

An matrix is diagonalizable iff it has linearly independent eigenvectors.

Equivalent conditions:

• has distinct eigenvalues → automatically diagonalizable
• Every eigenvalue’s geometric multiplicity equals its algebraic multiplicity (see Characteristic Polynomial)
• has a basis consisting of eigenvectors of (an eigenbasis)

Some matrices are not diagonalizable — e.g., has only one eigenvalue () of algebraic multiplicity but geometric multiplicity . For non-diagonalizable matrices, see Jordan Canonical Form.

93.2. How to diagonalize

1. Find the eigenvalues from .
2. For each , find a basis for — the eigenvectors.
3. Assemble eigenvectors into columns of , eigenvalues into the diagonal of (in matching order).
4. Verify by computing .

93.3. Why it matters: power and exponential

In diagonal form, matrix arithmetic is trivial:

with just powering each diagonal entry. This makes:

• for large tractable (Markov chains, dynamics)
• Matrix exponential (linear ODE solutions)
• Solving linear recurrences in closed form

93.4. Spectral decomposition (real symmetric case)

If is real and symmetric, then it’s always diagonalizable and can be done with an orthogonal :

This is the Spectral Theorem — the gold standard of diagonalization.

93.5. Failure: non-diagonalizable matrices

has only the eigenvalue (with algebraic multiplicity ) and only one independent eigenvector . Not diagonalizable — the best you can do is the Jordan form.

93.6. See also

• Eigenvectors & Eigenvalues
• Characteristic Polynomial
• Spectral Theorem — symmetric case
• Jordan Canonical Form — non-diagonalizable case
• Change of Basis — what does geometrically