Math 102  Homework 8 (selected problems)
David Lipshutz
Problem 1.
(Strang, 5.5: #14)
In the list below, which classes of matrices contain
A
and which contain
B
?
A
=
0
1
0
0
0
0
1
0
0
0
0
1
1
0
0
0
and
B
=
1
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Orthogonal
,
invertible
,
projection
,
permutation
,
Hermitian
,
rank
1,
diagonalizable
,
Markov
.
Find the eigenvalues of
A
and
B
.
Proof.
A
is orthogonal, invertible, not projection (since
A
6
=
A
2
), permutation, not Hermitian
(since
A
6
=
A
T
), diagonalizable and Markov. The characteristic polynomial for
A
is
p
(
λ
) =
λ
4

1, so the eigenvalues of
A
are
±
1 and
±
i
.
B
is projection (onto the space spanned by
(1
,
1
,
1
,
1)), Hermitian, rank1, diagonalizable, Markov. The eigenvalues of
B
are 1 since it is
a Markov matrix and the rest 0 since it is rank1.
Problem 2.
(Strang, 5.5: #16)
Write one significant fact about the eigenvalues of each of the following.
(a) A real symmetric matrix.
(b) A stable matrix: all solutions to
du/dt
=
Au
approach zero.
(c) An orthogonal matrix.
(d) A Markov matrix.
(e) A defective matrix (nondiagonalizable).
(f) A singular matrix.
Proof.
(a) Every eigenvalue is real.
(b) All eigenvalues have norm less than 1.
(c) Eigenvalues all have norm equal to 1.
(d) 1 is an eigenvalue of the matrix and the other eigenvalues are less than 1.
(e) The matrix has repeated eigenvalues.
(f) 0 is an eigenvalue of the matrix.
1
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Problem 3.
(Strang, 5.5: #18)
Show that a unitary matrix has

det
U

= 1, but possibly det
U
is different from det
U
H
.
Describe all 2 by 2 matrices that are unitary.
Proof.
If
λ
1
, . . . , λ
n
are the eigenvalues of
U
, then

det
U

=

λ
1
· · ·
λ
n

=

λ
1
 · · · 
λ
n

= 1.
Suppose
U
=
"
1
0
0
i
#
then
U
H
=
"
1
0
0

i
#
so det
U
=
i
and det
U
H
=

i
.
Suppose
U
=
"
r
1
e
iθ
1
r
3
e
iθ
3
r
2
e
iθ
2
r
4
e
iθ
4
#
is unitary.
Then
U
has
orthonormal columns, so
r
2
1
+
r
2
2
= 1 =
r
2
3
+
r
2
4
.
Let
r
1
= sin
ϕ
1
then
r
2
= cos
ϕ
1
, and if
r
3
= cos
ϕ
2
then
r
4
= sin
ϕ
2
where 0
≤
ϕ
1
, ϕ
2
≤
π
2
.
By the orthogonality of the column
vectors, sin
ϕ
1
cos
ϕ
2
e
i
(
θ
1
+
θ
3
)
+ cos
ϕ
1
sin
ϕ
2
e
i
(
θ
2
+
θ
4
)
= 0.
This implies that
ϕ
2
=
ϕ
1
and
e
θ
1
+
θ
3
=

e
θ
2
+
θ
4
⇒
θ
1
+
θ
3
=
θ
2
+
θ
4
+
π
. So
U
=
"
sin
ϕe
iθ
1
cos
ϕe
iθ
3
cos
ϕe
iθ
2
sin
ϕe
i
(
θ
1
+
θ
3

θ
2

π
)
#
or
"
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 Fall '08
 SZYPOWSK
 Math, Linear Algebra, Matrices, Orthogonal matrix, Strang

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