Diagonalizable

Diagonalizable refers to a square matrix or linear operator that can be represented as a diagonal matrix in some basis. A matrix A over a field F is diagonalizable over F if there exists an invertible P in F^{n×n} such that P^{-1}AP is diagonal. Equivalently, A is diagonalizable if there exists a basis consisting entirely of eigenvectors of A; in that case A can be written as A = PDP^{-1}, where D is diagonal and contains the eigenvalues of A on its diagonal.

Several equivalent criteria are commonly used. If A has n distinct eigenvalues in F, then A is diagonalizable. A diagonalizable matrix has a full set of linearly independent eigenvectors, meaning the sum of the dimensions of its eigenspaces equals n. The minimal polynomial m_A(x) of A over F must split into distinct linear factors (no repeated roots); equivalently, A has no Jordan blocks of size greater than 1 in its Jordan form over an algebraic closure of F. Over fields where the characteristic polynomial does not split, diagonalizability is considered with respect to that field; in such cases A may be diagonalizable over an extension field.

Examples illustrate the concept. The identity matrix is diagonalizable. Any diagonal matrix is diagonalizable. If a matrix has n distinct eigenvalues, it is diagonalizable. A single Jordan block [λ 1; 0 λ] is not diagonalizable. A rotation matrix in the plane is diagonalizable over the complex numbers but not over the real numbers when the angle is not a multiple of π. Notable special cases include real symmetric matrices and complex Hermitian matrices, which are diagonalizable by orthogonal (or unitary) transformations, with real eigenvalues in the real case.