Let V be a finite-dimensional linear spacePlanetmathPlanetmath over a field K, and T:VV a linear transformation. To diagonalize T is to find a basis of V that consists of eigenvectorsMathworldPlanetmathPlanetmathPlanetmath. The transformation is called diagonalizable if such a basis exists. The choice of terminology reflects the fact that the matrix of a linear transformation relative to a given basis is diagonal if and only if that basis consists of eigenvectors.

Next, we give necessary and sufficient conditions for T to be diagonalizable. For λK set


It isn’t hard to show that Eλ is a subspacePlanetmathPlanetmathPlanetmath of V, and that this subspace is non-trivial if and only if λ is an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of T. In that case, Eλ is called the eigenspaceMathworldPlanetmath associated to λ.

Proposition 1

A transformation is diagonalizable if and only if


where the sum is taken over all eigenvalues of the transformation.

The Matrix Approach.

As was already mentioned, the term “diagonalize” comes from a matrix-based perspective. Let M be a matrix representationPlanetmathPlanetmath (http://planetmath.org/matrix) of T relative to some basis B. Let


be a matrix whose column vectorsMathworldPlanetmath are eigenvectors expressed relative to B. Thus,


where λi is the eigenvalue associated to vi. The above n equations are more succinctly as the matrix equation


where D is the diagonal matrixMathworldPlanetmath with λi in the i-th position. Now the eigenvectors in question form a basis, if and only if P is invertiblePlanetmathPlanetmathPlanetmath. In that case, we may write

M=PDP-1. (1)

Thus in the matrix-based approach, to “diagonalize” a matrix M is to find an invertible matrix P and a diagonal matrix D such that equation (1) is satisfied.


There are two fundamental reasons why a transformation T can fail to be diagonalizable.

  1. 1.

    The characteristic polynomialMathworldPlanetmathPlanetmath of T does not factor into linear factors over K.

  2. 2.

    There exists an eigenvalue λ, such that the kernel of (T-λI)2 is strictly greater than the kernel of (T-λI). Equivalently, there exists an invariant subspace where T acts as a nilpotent transformation plus some multipleMathworldPlanetmathPlanetmath of the identityPlanetmathPlanetmathPlanetmath. Such subspaces manifest as non-trivial Jordan blocksMathworldPlanetmath in the Jordan canonical form of the transformation.

Title diagonalization
Canonical name Diagonalization
Date of creation 2013-03-22 12:19:49
Last modified on 2013-03-22 12:19:49
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 16
Author rmilson (146)
Entry type Definition
Classification msc 15-00
Related topic Eigenvector
Related topic DiagonalMatrix
Defines diagonalise
Defines diagonalize
Defines diagonalisation
Defines diagonalization