2010
Homework ECE210A
Fall 4
Due date: November 1, 2010.
1. Reading assigment. Read chapter 4 of the class notes.
2. If X and Y are full column rank matrices of rank p, show that the rank
of XY T is equal to p.
3. Show that if A is a rank p matrix then you can nd two full column rank
matrices X and Y of rank p such that A = XY T .
4. If X and Y are two full column rank matrices of rank p, nd all matrices
A for which R(X ) = R(A) and R(AT ) = R(Y ).
5. Let X and Y be two full column rank real matrices. What are the conditions (if any) on X and Y such that there exists a real matrix A such that
AX = Y and Y T A = X T ? Find all such A when the conditions are satised. How many linearly independent equations were there? What was
the dimensionality of the and left right nullspaces? Did the rank-nullity
theorem hold?
6. Suppose A Rmn , y Rm , and 0 = s Rn . Show that E = (y
As)sT /sT s has the smallest 2-norm of all m n matrices E that satisfy
(A + E )s = y .
7. Let X and Y be two full column rank matrices. We know from problem
5 that there are many matrices A which satisfy the pair of equations
AX
=
Y TA =
Y,
XT .
From the set of all such matrices A, nd the one that has the least Frobenius norm.
8. Let M be a real symmetric n n matrix, and let LM be a linear operator
from Rnn to Rnn , dened by the equation
LM (A) =
AT M + M A
.
2
Dene the matrix (or operator) 2-norm of LM (denoted by LM
LM
2
= max
A
F =1
LM (A)
Show that
LM
Extra credit: Show that LM
2
2
M
=M
1
2.
F.
F.
2)
by