5.3 - Diagonalization

A matrix is diagonalizable if:

APDP

where:

• matrix of eigenvectors

• diagonal matrix of eigenvalues
  

Main Theorem:

An matrix is diagonalizable it has linearly independent eigenvectors.

Fast Diagonalization Steps:

1. Find eigenvalues by solving:

2. Find eigenvectors for each eigenvalue

3. If total # of independent eigenvectors n diagnosable

4. Form

P [v1 v2...], D diag(λ​,λ)

Key Facts

• Distinct eigenvalues automatically diagonalizable

• Matrix may still be diagonalizable with repeated eigenvalues, but only if:​

  geom mult alg mult

• If fewer eigenvectors than dimensions, matrix is not diagonalizable

• P,D are not unique (you may reorder eigenvalues/eigenvectors)

Using Diagonalization

To compute powers:

Just RMB:

• Do I have eigenvectors? If yes, diagonalizable.

• Are eigenvalues all distinct? Then definitely diagonalizable.

• Repeated eigenvalue? Check if you get enough eigenvectors

• Need to compute ? Use