Math 818: Problem set 3,
due February 4
Instructions: Write up any four of the following problems (but
you should try all of them, and
in the end you should be sure you know how to do all of them).
Your write ups are due Friday, February 4, 2005.
Each problem is worth 10 points, 9 points for correctness
and 1 point for communication. (Your goal is not only
to give correct answers but to communicate
your ideas well. Make sure you use good English,
so proofread your solutions.
Once you finish a solution, you
should restructure awkward sentences, and strike out
anything that is not needed in your approach to the problem.)
- Let A be an m x m matrix with entries in a field F.
Prove A is skew symmetric if and only if < , >A
is a skew symmetric form on Fm.
- Let W be a subspace of an n dimensional
F-vector space V, where F is a
field. Let < , > be a skew symmetric form on V.
Define a form < , >' on W by
< v,w >' = < v,w > for all v and w in W.
Let N be the nullspace of < , > (i.e., the
subspace N of V, defined by the condition that
x is in N if and only if < v,x > = 0 for all v in V).
- (a) Show that < , >' is a skew symmetric form on W.
- (b) If the intersection of W and N is 0 and if W + N = V,
then show that < , >' is nondegenerate.
- (c) Show that there exists a basis B of V, such that
with respect to this basis the matrix A for < , >
is a block diagonal matrix with two blocks. The block in the upper
left corner of A is an r x r zero matrix, and the block in the
lower right corner of A is J2m, where r is the dimension
of N and r + 2m = n. [Hint: Start with a basis of N. Extend it
to a basis of V. Let W be the span of the basis vectors not in
N. Now use Theorem 8.5 to get a better basis for W.]
- Let U be the 3 x 3 upper triangular matrix with every entry on the
diagonal being 0 and every entry above the diagonal being 1.
Let A = U - Ut. Note that A is a skew symmetric matrix.
Find an invertible matrix P such that PtAP has the form
described in the previous problem.
- Let A be a complex normal nxn matrix.
- (a) Prove that A is hermitian if and only if all its eigenvalues are real.
- (b) Prove that A is unitary if and only if all its eigenvalues have norm 1.
- Prove or give an explicit counterexample: the nonzero roots
of the characteristic polynomial of a real nxn skew symmetric
matrix A are all purely imaginary. [Hint: Look at iA.]