Diagonalizables

Diagonalizables are square matrices or linear operators that can be transformed into a diagonal form by a similarity transformation. Equivalently, a matrix A is diagonalizable if there exists an invertible matrix P such that P^{-1}AP is diagonal. In this case the columns of P can be chosen as a basis of eigenvectors of A, and the diagonal entries of the diagonal form are the corresponding eigenvalues.

Several equivalent conditions characterize diagonalizability. A matrix is diagonalizable over a field F if and only if the vector space has a basis consisting entirely of eigenvectors of the matrix. This occurs precisely when, for each eigenvalue λ, the geometric multiplicity (the dimension of the eigenspace of λ) equals the algebraic multiplicity (the multiplicity of λ as a root of the characteristic polynomial), and the sum of these dimensions equals the size of the matrix. Another common criterion is that the minimal polynomial of A splits into distinct linear factors over F (no repeated roots). In particular, a matrix with n distinct eigenvalues is always diagonalizable.

Examples and caveats. Any identity matrix is diagonalizable. A matrix with n distinct eigenvalues is diagonalizable. A non-diagonalizable example is a Jordan block of size greater than 1 for a repeated eigenvalue. Diagonalizability over the real numbers may fail if eigenvalues are nonreal, while over the complex numbers it always holds under the distinct-eigenvalues condition. Normal matrices are unitarily diagonalizable, a stronger property than mere diagonalizability.

Applications include simplifying powers and functions of matrices, solving systems of linear differential equations, and revealing long-term dynamics in linear models.