Matrix Diagonalization & Similarity Transform

Compute the diagonalization1 of a square matrix \(A\), finding matrices \(P\), \(D\), \(P^{-1}\) such that \(P^{-1}AP = D\)2.
Matrix entries can be real, complex, fractions, or expressions like \(\pi\), \(e\), \(\sqrt{2}\), etc.

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 similar3, 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 eigenvalues4

Applications:

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

Enter the matrix size n and click Generate Matrix to create an n×n input grid.

1 — Set Up
Loads a pre-filled 4×4 matrix — ready to diagonalize.
2 — Compute
Finds eigenvalues and eigenvectors, then diagonalizes the matrix as A = PDP⁻¹.
Resets all dimensions, matrix entries, and results.