MATH 235/W08: Orthogonal Diagonalization,
Symmetric & Complex Matrices, Assignment 8
Hand in questions 1,3,5,7,9,11,13 by 9:30 am on Wednesday April 2, 2008.
Contents
1 Properties of Symmetric/Hermitian/Normal Matrices***
2
2 More on Hermitian/Unitary Matrices
2
3 Hermitian, Orthogonal Projections***
2
4 Hermitian and SkewHermitian Parts
2
5 Quadratic Forms***
2
6 Normal Matrices
3
7 Orthogonal Diagonalization***
3
8 Eigenspaces
3
9 Unitary Diagonalization***
3
10 Symmetric Square Root
3
11 Orthogonal Eigenvectors***
4
12 Common Eigenpairs
4
13 MATLAB***
5
13.1 Colliding Eigenvalues***
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
13.2 Equation of an Orbit***
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1
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1
Properties of Symmetric/Hermitian/Normal Matri
ces***
A (complex)
normal matrix
is defined by
A
*
A
=
AA
*
; it has orthogonal eigenvectors.
A
skew
Hermitian matrix
is defined by
A
*
=

A
.
1. Why is every skewHermitian matrix normal?
2. Why is every unitary matrix normal?
3. For what values of
a,d
is the 2
×
2 matrix
parenleftbigg
a
1

1
d
parenrightbigg
normal?
2
More on Hermitian/Unitary Matrices
1. Let
A,B
be
n
×
n
matrices and suppose
B
=
A

1
A
T
and
B
is symmetric. Prove that
A
2
is
symmetric.
2. Suppose
C
is a real
n
×
n
matrix such that
C
is symmetric and
C
2
=
C
and let
D
=
I
n

2
C
with
I
n
denoting the
n
×
n
identity matrix. Prove that
D
is symmetric and orthogonal.
3. Find all complex 2
×
2 matrices
A
= [
a
ij
] which are both unitary and Hermitian, and have
a
11
= 1
/
2.
3
Hermitian, Orthogonal Projections***
Let
Z
be an
m
×
n
complex matrix such that
Z
*
Z
=
I
n
where
I
n
denotes the
n
×
n
identity matrix.
1. Show that
H
=
ZZ
*
is Hermitian and satisfies
H
2
=
H
.
2. Show that
U
=
I
n

2
ZZ
*
is both unitary and Hermitian.
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 Winter '08
 CELMIN
 Linear Algebra, Algebra, Matrices, Orthogonal matrix, Normal matrix, Symmetric matrix, Hermitian matrix

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