Diagonalizing a Matrix

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Diagonalizing A = PDP^ -1 A complete worked process: eigenvalues, eigenvectors, building P and D , then verifying A = PDP^ -1 . Diagonalize the matrix A below. Find an invertible P and a diagonal D with A = PDP^ -1 , then verify the factorization by reconstructing A . Along each eigenvector direction, multiplying by A just stretches the vector by its eigenvalue — the direction never turns. Those three special directions become the columns of P , and the stretch factors fill the diagonal of D . Nice work — you diagonalized A from start to finish and confirmed A = PDP^ -1 . Eigenvalues from : for a triangular matrix the determinant is the product of the diagonal entries, so the eigenvalues are just 1, 3, 2 — read straight off the diagonal. Eigenvectors from the null space: for each , solve to get the direction A only scales. P holds the eigenvectors as columns; D holds the eigenvalues: the column order of P must match the diagonal order of D . The factorization: a matrix with n independent eigenvectors satisfies A = PDP^ -1 , equivalently P^ -1 AP = D . Diagonalization makes hard matrix work easy: powers become A^n = PD^nP^ -1 , and the diagonal form exposes how the transformation acts along each eigen-direction.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.