If $A$ is diagonalizable, we find matrices $P$ (whose columns are eigenvectors), $D$ (diagonal matrix of eigenvalues), and $P^{-1}$ such that:

$ A = P D P^{-1} \quad \text{or equivalently} \quad D = P^{-1} A P $

Similarity Transformation:

Matrices $A$ and $D$ are similar, meaning they represent the same linear transformation in different bases.

When is $A$ diagonalizable?

• If $A$ has $n$ linearly independent eigenvectors (always true if eigenvalues are distinct)
• If $A$ is symmetric (or Hermitian), it's always orthogonally diagonalizable
• Algebraic multiplicity = geometric multiplicity for all eigenvalues

Applications:

• Computing matrix powers: $A^k = PD^kP^{-1}$
• Solving systems of differential equations
• Principal component analysis (PCA)