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Title: Hermitian matrices and unitary diagonalization
Description: Linear algebra course
Description: Linear algebra course
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Hermitian Matrices
and Unitary
Diagonalization
9
...
1, we saw that every real symmetric matrix is orthogonally diagonalizable
...
First, we observe that if A is a real symmetric matrix, then the condition AT = A
is equivalent to A = A
...
*
Definition
Hermitian Matrix
Ann xn matrix A with complex entries is called Hermitian if A
if A= Ar
...
BT,so
B
...
...
i
-i * cT, soc is not Hermitian
...
j the ij-th entry must be the complex conjugate of the ji-th
entry
...
Aw> =
Proof: If
A is Hermitian, then we get
(z,Aw) = t Aw= zTA� = t AT�= (Azl� = (Az, w)
If (z,Aw) =
(Az,w) for all z, w E e11, then we have
Since this is valid for all
t,w E
en' we have that A=
AT
...
Remark
A linear operator L : V � V is called Hermitian if (x, L(j)) = (L(x),y) for all
x,y E V
...
Hermitian linear operators play an
important role in quantum mechanics
...
Then
(1) All eigenvalues of A are real
...
Proof: To prove (1), suppose that ;l is an eigenvalue of A with corresponding unit
eigenvector t
...
Then
ing eigenvectors
(t1,At2)
=
(t1,A2t2)
=
(t,AZ)
(At,Z)
and
�
A2(l1,t2)
A is Hermitian, we get ;l2(t1,l2)
T hus,
=
,i
;l must be real
...
Thus, since
A1
*
=
A1(l1,l2)
A2,we must have
=
From this result, we expect to get something very similar to the principal axis
theorem for Hermitian matrices
...
EXAMPLE2
Let
A
=
[ :
...
Solution: We have A*
Verify that
=
A, so A
A is
Hermitian and diagonalize
C(;l)
Then the characteristic polynomial is
or
;l
1
...
Consider
=
,12 - 5,i + 4
=
A
...
(,1- 4)(,1- 1 ) ,so ;l
=
4
=
=
4,
A_ ,if
=
[
1
-2
...
If
I ] [
[ �]
+
i
_
2
=
1_ i
1
-2
0
;l
1
0
=
1
+
0
]
i
...
Hence, A is diagonalized to
[� �]
Observe in Example 2 that since the columns of Qare orthogonal, we can make
Qunitary by normalizing the columns
...
We can prove that we can do this for any Hermitian
matrix
...
Then there exist a unitary matrix U
and a diagonal matrix D such that U*AU = D
...
You are asked to prove the theorem
as Problem DS
...
If A and Bare matrices such that B= U*AU for some U, we say that A and B
are unitarily similar
...
2
...
That is, there exist matrices that are unitarily
diagonalizable but not Hermitian
...
1
i
onalizable
...
PROBLEMS 9
...
L
+i
...
/2
...
(a) A=
(b) B
[
=
[
24
...
-
i
]
]
(d) F
=
[ V:+
l
1
l
0
-
i
i
1 +i
0
1 +i
Homework Problems
Bl For each of the following matrices,
(i) Determine whether it is Hermitian
...
(a) A
(b) B
=
=
2
-f2
...
5
+i
+i
Y2 +
...
/3
_
i
i
]
...
-�]
i
3
Conceptual Problems
Dl Suppose that A and B are
n x n
Hermitian matri
(b) What can you say about
ces and that A is invertible
...
(c) What can you say about the form of a
(c) A-1
D2 Prove (without appealing to diagonalization) that if
A is Hermitian, then detA is real
...
(a) AB
D3 A general
a,
2 2
� ci
b
x
b
Hermjtian matrix can be written as
i
: c ]
...
Prove
that a linear operator L : V -t V is Hermitian
( (1, L(j))
(L(x), y)) if and only if its matrix with
respect to any orthonormal basis of V is a Hermi
=
tian matrix
...
unitary as well as Hermitian
...
How does the complex conjugate relate to division of
complex numbers? How does it relate to the length
of a complex number? (Section 9
...
Ex
plain how to convert a complex number from stan
dard form to polar form
...
1)
3 List some of the similarities and some of the differ
ences between complex vector spaces and real vector
spaces
...
(Section 9
...
4)
5 Define the real canonical form of a real matrix A
...
4)
6 Discuss the standard inner product in C"
...
5)
7 Define the conjugate transpose of a matrix
...
(Section 9
...
6)
a
Hermitian
matrix
...
Z1
...
Use polar form to
E2 Use polar form to determine all values of (i)112
...
(a) 2it + (1 + i)V
(b) it
(c) (it, v)
Cd)
(e) llVll
(f) proja v
E4 Let A=
[� 1
-
13
4 ·
(a) Determine a diagonal matrix similar to A over
C and give a diagonalizing matrix P
...
ES Prove that U
matrix
...
V3
� ki
[1
l
-
1
i
l
-i
is a unitary
-l+i
(a) Determine k such that A is Hermitian
...
Verify that the eigenvectors are
orthogonal
...
Prove that A has a square
that every n x n matrix A is unitarily triangulariz
root
...
(This is called Schur's Theorem
...
(Hint: Suppose that U diago
nalizes A to D so that U* AU= D
...
)
F4 A matrix is said to be normal if A*A= AA*
...
F2 (a) If A is any n x n matrix, prove that A*A
(b) Use (a) to show that if A is normal, then A is
is Hermitian and has all non-negative real
unitarily similar to an upper triangular matrix
eigenvalues
...
F3 A matrix A is said to be unitarily triangularizable
if there exists a unitary matrix U and an upper tri
angular matrix T such that U*AU = T
...
(c) Prove that every upper triangular normal matrix
is diagonal and hence conclude that every nor
mal matrix is unitarily diagonalizable
...
)
Title: Hermitian matrices and unitary diagonalization
Description: Linear algebra course
Description: Linear algebra course