Math 421/621

Spring 2000

Exam 1

1. Assume that A and B are similar matrices (over some field, F):

a) Show that is an eigenvalue of A if and only if it is an eigenvalue of B.

b) Show that det(A)=det(B).

c) (G) Show that the det(A) (hence det(B)) is the product of the eigenvalues of A (you may assume that A is an nxn matrix and that you can actually find all n of the eigenvalues (counting multiplicity) of A in the field F).

2. Let A be a linear transformation with an eigenvalue, eigenvector pair given by . Show that is an eigenvalue, eigenvector pair for
3. Let A be a linear transformation of the vector space , and let be an eigenvalue of A. Show that is a subspace of (This is called the eigenspace for
4. An nxn matrix is called a diagonal matrix if it is of the form

A diagonal matrix is called a scalar matrix if

a) If A is a scalar matrix, show that it is only similar to itself (in particular, if a matrix is similar to a scalar matrix, then it is a scalar matrix).

b) If a matrix is similar to a diagonal matrix, is it necessarily diagonal?

5. (G) Let A be an nxn matrix over a field, F. A is said to be diagonalizable if it is similar to a diagonal matrix. Show that A is diagonalizable if and only if the eigenvectors of A form a basis for
6. A matrix is said to be nilpotent if is the 0-matrix for some Give an example of a nonzero nilpotent matrix and show that the eigenvalues of any nilpotent matrix (over a field) are all 0.