For , the eigenvector satisfies for some eigenvalue .
The algebraic multiplicity of is the exponent of in the characteristic polynomial of , .
The geometric multiplicity of is the dimension of the eigenspace of , .
Let the algebraic multiplicity and geometric multiplicity distinct eigenvalues be and respectively. Then:
.
If , then is diagonalizable. (i.e. independent eigenvectors for ). Hence, a basis of eigenvectors . Then, .
1. Generalized Eigenvectors
A non zero vector is a generalized eigenvector of rank with associated eigenvalue if:
Thus, any eigenvector associated with is a generalized eigenvector of rank .
In summary, the generalized eigenvector of rank is a vector that is not an eigenvector but has the property that when you apply to it times, you get the zero vector. The generalized eigenvectors are used to find the Jordan form of a matrix.
2. Jordan Normal Form (JNF)
A matrix is in jordan normal form if it is in the following form:
The jordan block of , , is a square matrix of size with on the diagonal and on the super-diagonal, where is the algebraic multiplicity of .
Suppose we have is represented by . If is diagonalizable, then going to the basis obtained by all its set of linearly independent eigenvectors, we get . This is the diagonal form of . Then, will be represented by a diagonal matrix in the basis .
If for eigenvalue of , , then the matrix is not diagonalizable, but can be put into jordan normal form. Instead, we say is a generalized eigenvector of if for . In general, .