Chapter 4

1. Chapter 4 Section 4.5 Eigenvectors and Eigenspaces

2. Eigenspaces and Geometric Multiplicity If A is a matrix and is an eigenvalue of A then we can define the following two concepts. The eigenspace of (denoted ) is the null space of the matrix . The geometric multiplicity of is the dimension of the eigenspace of . Geometric Multiplicity of : Dimension of The eigenspace of a matrix is obviously a subspace since it is the null space of a matrix. All of the vectors v in eigenspace are eigenvectors corresponding to . Example For the matrix A to the right find the eigenvalues, a basis for each eigenspace and the algebraic and geometric multiplicities of the eigenvalues. The matrix A is upper triangular: Eigenvalues: 5,7 row reduces to , vector form: The eigenspace has dimension 1.

3. row reduces to , vector form: The eigenspace has dimension 2. For , a basis for is , algebraic multiplicity =1, geometric multiplicity is =1. For , a basis for is, algebraic multiplicity =3, geometric multiplicity is =2. Theorem If A is a matrix and is an eigenvalue of A then the algebraic and geometric multiplicities of the eigenvalue will always conform to the following inequality: The multiplicities are not necessarily equal. This tells us something about the matrix.

4. Defective Matrices A matrix A is called defective if there are one or more of its eigenvalues such that the geometric multiplicity is not equal to the algebraic multiplicity. Matrix A is defective if there is an eigenvalue such that: To determine if a matrix A is defective or not can be difficult. The method to determine the geometric multiplicity of an eigenvalue is to find a basis for or at least find the nullity of . This is often very involved, you need to row reduce a matrix then write the vector form of the solution if you are finding a basis. There are a few things that will cut down on the number of times this needs to be done. Theorem If A is a matrix with eigenvalue whose algebraic multiplicity is 1 then the geometric multiplicity of is equal to 1 also. This follows from: This says that an eigenvalue with algebraic multiplicity 1 is never the reason a matrix is defective. Check only the eigenvalues whose algebraic multiplicity is greater than 1. Theorem If A is a matrix with n distinct (different) eigenvalues then A is not defective. Since each eigenvalue is different each eigenvalue has algebraic multiplicity 1. Because there are n of them no complex eigenvalue exists to have algebraic multiplicity greater than 1.

5. Example For the matrix A to the right find its eigenvalues, a basis for each of its eigenspaces, the algebraic and geometric multiplicities and if it is defective or not. To get the characteristic polynomial try a 1st row expansion. Simplify the polynomial to factor it. The characteristic polynomial is: Eigenvalues are: -2,1 The possible roots are: Calculate , 1 is a root. Use long division to get We get The eigenvalue 1 has algebraic multiplicity of 1 so it will not cause the matrix to be defective. We need to check the eigenvalue -2 since that has algebraic multiplicity of 2.

6. row reduces to , vector form: The eigenspace has dimension 2. This means that the geometric multiplicity of the eigenvalue -2 is 2 which is equal to its algebraic multiplicity. This matrix is not defective. The question also asked for a basis for , we know it has dimension 1 but we will still need to row reduce the matrix to find a basis. row reduces to , vector form: The eigenspace has dimension 2. For , a basis of is , algebraic multiplicity =1, geometric multiplicity is =1. For , a basis of is, algebraic multiplicity =2, geometric multiplicity is =2. This matrix is not defective since all algebraic multiplicities are equal to all geometric multiplicities.

7. Example For the matrix A given to the right find the eigenvalues, algebraic and geometric multiplicity of each eigenvalue and determine if the matrix is defective or not. The matrix is upper triangular so the characteristic polynomial for this matrix is given by: The eigenvalues are: -2, 2, 3 and 6 which are all distinct. That means that all of the algebraic and geometric multiplicities are 1 and the matrix is not defective. Theorem If A is a matrix and is an eigenvalue of A then the geometric multiplicity of the eigenvalue . The geometric multiplicity of is the dimension of the null space of , which is the nullity of . Apply the rank – nullity equation to this matrix to get: Theorem If a matrix A is diagonal then the matrix A is not defective. If A is diagonal then the eigenvalues are the diagonal entries. If is a diagonal entry then the geometric multiplicity of is equal to which is the number of zero rows in which is the number of times is on the diagonal which is the algebraic multiplicity.

8. Example Show any matrix A of the form to the right is not defective. If then the matrix is diagonal and hence not defective. Assume . Char polynomial: Apply the quadratic formula to get the roots. , Sincethenthis equation has two real roots: and These are the eigenvalues and they are distinct (different) so this matrix is not defective. This is the situation for a much more general result. Theorem If a matrix A is symmetric (i.e. ) then the matrix A is not defective.