Diagonalized

Diagonalized, in linear algebra, refers to a square matrix or linear operator that is similar to a diagonal matrix. A matrix A is diagonalizable over a field F if there exists an invertible matrix P with entries in F such that P^{-1} A P = D, where D is diagonal. Equivalently, there is a basis of F^n consisting of eigenvectors of A; in that basis the matrix representation is diagonal with the eigenvalues on the diagonal.

A key criterion is that A is diagonalizable if and only if A has n linearly independent eigenvectors, i.e., the sum of the dimensions of its eigenspaces is n. This occurs when the minimal polynomial of A splits into distinct linear factors over F (no repeated roots). Over the complex numbers, this is a common way to test diagonalizability; over the real numbers, a matrix with nonreal eigenvalues may be diagonalizable over C but not over R.

Computation proceeds by finding eigenvalues from det(A − λI) = 0, then solving (A − λI)x = 0 to obtain eigenvectors. If one can collect n independent eigenvectors, form P from these eigenvectors as columns; then D = P^{-1} A P is diagonal, with the corresponding eigenvalues on its diagonal.

Not all matrices are diagonalizable. For example, A = [[1,1],[0,1]] has a double eigenvalue 1 but only one line of eigenvectors, so it is not diagonalizable. In contrast, A = [[2,1],[0,3]] has eigenvalues 2 and 3 with independent eigenvectors and is diagonalizable.

Diagonalization is useful because it simplifies powers and functions of A: if A = P D P^{-1}, then A^k = P D^k P^{-1}.