Below are some basic properties of eigenspaces.
can be viewed as the kernel of the linear transformation . As a result, is a subspace of .
In fact, if is the sum of eigenspaces corresponding to eigenvalues of other than , then .
From now on, we assume finite-dimensional.
Let be the set of all eigenvalues of and let . We have the following properties:
If is the algebraic multiplicity of , then .
In other words, if splits over , then is diagonalizable iff .
For example, let be given by . Using the standard basis, is represented by the matrix
From this matrix, it is easy to see that is the characteristic polynomial of and is the only eigenvalue of with . Also, it is not hard to see that only when . So is a one-dimensional subspace of generated by . As a result, is not diagonalizable.
|Date of creation||2013-03-22 17:23:07|
|Last modified on||2013-03-22 17:23:07|
|Last modified by||CWoo (3771)|