Math 508
Solution to Problem 5 (a) on Page 232
(7/29/03)
5. Unitarily diagonalize the matrix
211
A = 1 2 1 .
112
Solution. First nd the eigenvalues of A. The characteristic polynomial of A is:
|I A| = 3 62 + 9 4 = (3 62 + 5) + (4 4) = (2 6 + 5) + 4( 1)
=
Math 508
Solution to Problem 1 (c) on Page 232
1 (c). Find a unitary
2 4
A = 1 2
3 6
(7/29/03)
matrix U such that U H AU is upper triangular.
2
1 .
3
Solution. We use the Schur decomposition process in two steps to nd U and T .
Step 1. Find U1 . We rst nd
Math 508
1. Let A =
Solutions to Quiz #5
4 1
. Find maxcfw_ Ax
05
:
x
= 1 and mincfw_ Ax
Solution. Note that maxcfw_ Ax : x = 1 = A
151
1/ A1 , and A1 = 20
. It follows that
04
maxcfw_ Ax
:
x
mincfw_ Ax
:
x
,
mincfw_ Ax
:
x
:
x
= 1.
= 1 =
= 1 = maxcfw_|4|
Math 508
Solutions to Quiz #2
(6/9/03)
1. (a) Find a basis for the spanS , where S consists of the following six matrices:
A1 =
11
,
10
A4 =
01
,
1 1
A2 =
A5 =
11
,
11
A3 =
1 1
,
10
1 1
,
01
A6 =
1 1
.
1 1
(b) Find the linear combinations for the rest of
Math 508
Solutions to Quiz #1
(6/4/03)
1. Use Gaussian or Gauss-Jordan eliminations to nd all the solutions of
x1 + x2 + 0x3 + x4 = 7
x1 + 0x2 + x3 + x4 = 4
0x1 + x2 + x3 + x4 = 10
Solution. We rst apply a series of row operations for the augmented matri
MA 508
Solutions to Midterm Test
Name: Shangbing Ai
June 30, 2003.
I. Determine whether the statement is true or false.
1. If two matrices A and B satisfy AB = 0, then either A = 0 or B = 0.
Solution: False. Counterexample: A = B =
01
= 0.
00
2. If two ma