Diagonalization

Diagonalization is the process of expressing a square matrix A as similar to a diagonal matrix. If there exists an invertible matrix P such that P^{-1}AP = D, where D is diagonal, then A is diagonalizable and D contains the eigenvalues of A on its diagonal. The columns of P are the corresponding eigenvectors of A.

A matrix is diagonalizable over a field F if and only if F^n has a basis consisting of eigenvectors of A, i.e., the geometric multiplicities of the eigenvalues sum to n. Equivalently, the minimal polynomial of A splits into distinct linear factors over F. In particular, a matrix with n distinct eigenvalues is diagonalizable. Over the real numbers, a real symmetric matrix is always diagonalizable by an orthogonal matrix: there exists an orthogonal Q with Q^T A Q = D, where D is real diagonal (the spectral theorem).

Computation typically follows these steps: find the eigenvalues by solving det(A − λI) = 0; for each eigenvalue λ, solve (A − λI)v = 0 to obtain eigenvectors; if n linearly independent eigenvectors can be found, form P with these eigenvectors as columns; then D = P^{-1}AP is diagonal and contains the eigenvalues on its diagonal. When a matrix is not diagonalizable, it is similar to a Jordan form rather than a diagonal matrix.

Diagonalization simplifies many computations, notably powers and functions of A, since A^k = P D^k P^{-1} whenever A = P D P^{-1} is diagonalizable.