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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
Title: Hermitian matrices and unitary diagonalization
Description: Linear algebra course
Description: Linear algebra course