diagonalization
Let be a finite-dimensional linear space over a field , and
a linear transformation. To diagonalize
is to find a basis of that consists of eigenvectors
. 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 to be diagonalizable. For set
It isn’t hard to show that is a subspace of , and that this subspace is
non-trivial if and only if is an eigenvalue
of . In that case, is called the eigenspace
associated to .
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
be a matrix representation (http://planetmath.org/matrix) of relative to some basis . Let
be a matrix whose column vectors are eigenvectors expressed relative
to . Thus,
where is the eigenvalue associated to . The above equations are more succinctly as the matrix equation
where is the diagonal matrix with in the -th
position. Now the eigenvectors in question form a basis, if and only
if is invertible
. In that case, we may write
(1) |
Thus in the matrix-based approach, to “diagonalize” a matrix is to find an invertible matrix and a diagonal matrix such that equation (1) is satisfied.
Subtleties.
There are two fundamental reasons why a transformation can fail to be diagonalizable.
-
1.
The characteristic polynomial
of does not factor into linear factors over .
-
2.
There exists an eigenvalue , such that the kernel of is strictly greater than the kernel of . Equivalently, there exists an invariant subspace where acts as a nilpotent transformation plus some multiple
of the identity
. Such subspaces manifest as non-trivial Jordan blocks
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 |