Matrix diagonalization is the process of transforming a square matrix A into a diagonal matrix D using a similarity transformation:

A = P D P⁻¹

Here, P is an invertible matrix whose columns are the eigenvectors of A, and D is a diagonal matrix containing the corresponding eigenvalues along its diagonal. This transformation simplifies many computations, such as raising a matrix to a power or finding its determinant.

Key condition: A matrix is diagonalizable if it has n linearly independent eigenvectors (for an n×n matrix). This is always true if all eigenvalues are distinct.

Procedure to Diagonalize a Matrix:

1. Find eigenvalues: Solve det(A - λI) = 0 to get λ values.

2. Find eigenvectors: For each λ, solve (A - λI)v = 0.

3. Check diagonalizability: Ensure the total number of independent eigenvectors equals n.

4. Form matrix P: Place eigenvectors as columns.

5. Form matrix D: Place eigenvalues on the diagonal in the same order as their eigenvectors in P.

6. Compute P⁻¹.

7. Verify: Ensure A = P D P⁻¹.