Math 421/621
Spring 2000
Exam 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).
- Let A be a linear
transformation with an eigenvalue, eigenvector pair given by
. Show that
is an eigenvalue, eigenvector pair for ![](./M421-621.EX1.S2000_files/image008.gif)
- 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 ![](./M421-621.EX1.S2000_files/image017.gif)
- An nxn matrix is called
a diagonal matrix if it is of the form
![](./M421-621.EX1.S2000_files/image019.gif)
A diagonal matrix is called a scalar matrix
if ![](./M421-621.EX1.S2000_files/image021.gif)
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?
- (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
![](./M421-621.EX1.S2000_files/image023.gif)
- 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.