Problem 1

(a). The Matrix Has an Eigenvalue . Find an Eigenvector for This Eigenvalue

(b). The Matrix Has an Eigenvector . Find the Eigenvalue for This Eigenvector

Problem 2

(a). The Matrix Has an Eigenvalue . Find an Eigenvector for This Eigenvalue

(b). The Matrix Has an Eigenvector . Find the Eigenvalue for This Eigenvector

Problem 3

The matrix has eigenvalue with an eigenspace of dimension, find a basis for the 2 eigenspace.

Problem 4

Let be the vector space of "smooth" functions, i.e., real-valued functions in the variable that have infinitely many derivatives at all points .

Let and be the linear transformations defined by the first derivative and the second derivative .

(a). Determine whether the Smooth Function is an Eigenvector of , if So, Give the Associated Eigenvalue

(b). Determine whether the Smooth Function is an Eigenvector of if So, Give the Associated Eigenvalue

Problem 5

Find the characteristic polynomial of the matrix

Problem 6

Find the eigenvalues of the matrix

Problem 7

For which value of does the matrix have one real eigenvalue of multiplicity 2.

Problem 8

The matrix has an eigenvalue of multiplicity 2 with corresponding eigenvector . Find and .

Problem 9

The matrix has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimension of its associated eigenspace.

The associated eigenspace for eigenvalue has dimensions 1.

Problem 10

Find the eigenvalues and eigenvectors of the matrix

Problem 11

Find the eigenvalues of the matrix .

Problem 2

The matrix has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and a basis for each eigenspace.