Problem 1

, , and are matrices

Check which of the following statements are true

• [ ] A. is diagonalizable if has distinct eigenvectors.
• [ ] B. If is invertible, then is diagonalizable.
• [x] C. If with being diagonal, then the nonzero columns of must be eigenvectors of .
• [ ] D. If is diagonalizable, then has distinct eigenvalues

Problem 2

Consider the matrix and let be a matrix similar to . Find all possible values of such that .

Problem 3

Let be a diagonalizable matrix who's eigenvalues are , and . If , , and are eigenvectors of corresponding to , , and respectively then factor into a product with diagonal, and use this factorization to find .

Problem 4

Find all possible values of , if any, for which the matrix

is not diagonalizable.

There is no value for which is not diagonalizable as the other elements in it's row and column are zero.

Problem 5

Let , find a matrix an a diagonal matrix such that .

Problem 6

Find a matrix such that , and are eigenvectors of with eigenvalues and respectively.

Problem 7

Let . Find two different diagonal matrices and the corresponding such that .

Problem 8

Let , find a formula for the entries of where is a positive integer.

Problem 9

Consider the sequence defined recursively by , , . We can use matrix diagonalization to find an explicit formula for

(a). Find the Matrix that Satisfies

(b). Find the Appropriate Exponent such that

(c). Find a Diagonal Matrix and an Invertible Matrix such that

(d). Find

(e). Find

(f). Use Parts (b). and (e). to Find

(g). Develop an Explicit Formula for Using part (b) and a Formula for

Problem 10

Find all the eigenvalues (real and complex) of the matrix .

Problem 11

Given , compute .

Problem 12

Find the eigenvalues and eigenvectors for

Problem 13

Suppose is a matrix with real entries that has a complex eigenvalue with corresponding eigenvector . Find another eigenvalue and eigenvector.