5.3 - Diagonalization

A matrix A is diagonalizable if:

A=PDP−1

where:

• P = matrix of eigenvectors

• D = diagonal matrix of eigenvalues
  

Main Theorem:

An n×n matrix is diagonalizable ↔ it has n linearly independent eigenvectors.

Fast Diagonalization Steps:

1. Find eigenvalues by solving: det*(A-λ*I)=0

2. Find eigenvectors for each eigenvalue

3. If total # of independent eigenvectors = n → diagnosable

4. Form

P = [v1 v2...], D = diag(λ1​,λ2,…)

Key Facts

• Distinct eigenvalues ⇒ automatically diagonalizable

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

  geom mult (λ)= alg mult (λ)

• If fewer eigenvectors than dimensions, matrix is not diagonalizable

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

Using Diagonalization

To compute powers:

Ak= PDk*p(-1)

Just RMB:

• Do I have n eigenvectors? If yes, diagonalizable.

• Are eigenvalues all distinct? Then definitely diagonalizable.

• Repeated eigenvalue? Check if you get enough eigenvectors

• Need to compute Ak? → Use Ak = PDk*p(-1)