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
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,
There are two fundamental reasons why a transformation can fail to be diagonalizable.
The characteristic polynomial of does not factor into linear factors over .
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.
|Date of creation||2013-03-22 12:19:49|
|Last modified on||2013-03-22 12:19:49|
|Last modified by||rmilson (146)|