The sum of all algebraic multiplicities for a matrix A canno

The sum of all algebraic multiplicities for a matrix A cannot exceed the sum of a geometric multiplicities for a matrix A.

Solution

Assume we know an eigenvalue .

The eigenspace of an eigenvalue is defined to be the linear space of all eigenvectors of A to the eigenvalue . The eigenspace is the kernel of A I_n.

Since we have computed the kernel a lot already, we know how to do that. The dimension of the eigenspace of is called the geometric multiplicity of . Remember that the multiplicity with which an eigenvalue appears is called the algebraic multiplicity of : The algebraic multiplicity is larger or equal than the geometric multiplicity. So, given statement is false.

Proof. Let be the eigenvalue. Assume it has geometric multiplicity m. If v1, . . . , v m is a basis of the eigenspace Eµ form the matrix S which contains these vectors in the first m columns. Fill the other columns arbitrarily. Now B = S 1AS has the property that the first m columns are µe1, .., µe m, where ei are the standard vectors. Because A and B are similar, they have the same eigenvalues. Since B has m eigenvalues also A has this property and the algebraic multiplicity is m. You can remember this with an analogy: the geometric mean ab of two numbers is smaller or equal to the algebraic mean (a + b)/2.