diagonalization (original) (raw)

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. LetM be a matrix representation (http://planetmath.org/matrix) of T relative to some basis B. Let

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

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

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

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.

Subtleties.

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