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Lecture 5 Hermitian Matrices


MATRIX ANALYSIS @ HITSZ
TIME: Autumn 2011 INSTRUCTOR: You-Hua Fan

Lecture 5: Hermitian Matrices
Reading assignment
?Section 4.1.1~4.1.5 ?Section 7.1.1~7.1.5 ?Section 7.2.1, 7.2,7 ?Section 7.3.5

1

Hermitian matrices form one of the most useful classes of square matrices. There are several very powerful facts about Hermitian matrices that have found universal application.
First the eigenvalues of Hermitian matrices are real.

Second, Hermitian matrices have a complete set of orthogonal eigenvectors, which makes them diagonalizable.
Third, these facts give a spectral representation for Hermitian matrices. In this lecture, without exception the underlying inner product is standard inner product and the underlying vector norm is Euclidean norm || · 2. ||
2

5.1 Diagonalizability of Hermitian Matrices

3

4

5

Let P1 ? [ x1 , u 2 , ? u n ], U 2 ? [ u 2 , ? u n ], then

? ?1 P AP 1 ? ? ?0
* 1

0 ? ?, A1 ?

A1 has the eigenvalue

s of ? 2 , ? , ? n

Just like the proof method for Schur’s theorem, we can find n orthonormal eigenvectors of A, and let A is unitarily equivalent to a diagonal matrix. This means for all eigenvalues.

6

( ii ) Let ? i and u i be the eigenvalue eigenvecto
*

s and pertaining

orthonorma

l

rs of A , then U ? [ u 1 , ? , u n ] is unitary .
* *

U AU ? U A [ u 1 , ? , u n ] ? U [ Au 1 , ? , Au n ] ? u 1* ? ? ? * ? ? ? ? [ ?1u 1 , ? , ? n u n ] ? [ u i ? j u j ] ? ?u * ? ? n? ? ?1 ? ? ? ? ? ? . ? ?n ? ?

?

7

A?

?

n j ?1

? ju ju (?
* j

?

n j ?1

Aj)

where A j is Hermiatian

.

Let U ? [ u 1 , ? , u n ] .

? ?1 A ?U ? ? ? ? ?

? ? U ? ?n ? ?

? ?1
*

? [u1 , ? , u n ]

? ? ? ?

?

? u 1* ? ? ?? ? ? ? ? ? * ? n ? ?u n ? ?? ?

? u 1* ? ? ? ? [ ?1u 1 , ? , ? n u n ] ? ? ? ? ?u * ? ? n?
* * * *

?

n

? ju ju j ?
*

j ?1

?

n

Aj

j ?1

A j ? (? ju ju j ) ? ? ju ju j ? ? ju ju j ? A j .
*

8

5.2 Skew-Hermitian Matrices and Properties of Quadratic Form
Definition 5.2.1. A matrix is Skew -Hermitian if A= –A*. Proposition 5.2.1 (Elementary Facts):

9

Theorem 5.2.1.

(1) If A is Hermitian.

x Ax ? x U ? Ux ? y ? y
* * * *

?

?
i

? i | y i | is real.
2

(2) If x Ax is real . Let A ? H ? iK , where H and K are Hermitian .

*

x Ax ? x ( H ? iK ) x ? x Hx ? ix Kx
* * * *

? 0 ? x Kx ?
*

??

i

| zi |

2

? ?i ? 0 ? K ? V diag ( ? i )V ? 0
*

10

Theorem 5.2.2.

Let A ? H ? iK , where
* *

H and K are Hermitian
* *

.

0 ? x Ax ? x ( H ? iK ) x ? x Hx ? ix Kx ? x Hx ? 0 , x Kx ? 0
* *

? H ? K ?0? A?0

11

( PD , PSD )

12

If A is Hermitian, then A is PD iff every eigenvalue of A is positive.

If B is PD, then any principal submatrix is PD and any principal minor is positive. In particular, diag (B) is positive and det (B) >0. 6. If A is PD then Ak is PD . 7. If A and B are PD then A+B is PD. 8. If A is PD then A=B*B, where B is invertible. 9. If B*B=0, then B=0.

13

Theorem 5.3.2. If A is positive (semi)definite. the square root of A is positive (semi) definite.

A ? U DU ? U diag ( ? i )U
* *

? U diag ( ? i ) diag ( ? i )U
*

? U diag ( ? i )UU diag ( ? i )U
* *

? B ( B ? U diag ( ? i )U )
2 *

B is PD ( PSD ) with eigenvalue

s of

?i .

14

Theorem 5.4.2. (Singular Value Decomposition)

15

are singular values of A.
16

AA is Hermitian,
?D 2 * * U AA U ? ? ?

*

then ? an unitary matrix U ? M
? ? , D ? diag (? 1 , ? , ? k ) 0?
m ,k

m

Let U ? [U 1 , U 2 ], U 1 ? M
* * 2 *

,U 2 ? M
*

m ,m ? k

U 1 AA U 1 ? D , U 2 AA U 2 ? 0 ? U 2 A ? 0
*

Let V1 ? A U 1 D
*
* *

?1

?M
*

n ,k

, then

U 1 AV 1 ? U 1 AA U 1 D
* ?1 * *

?1

? D D
2

?1

? D

V 1 V 1 ? D U 1 AA U 1 D

?1

? I

17

Let V ? [V1 , V 2 ] ? M
*

n

be unitary . then
* 2

0 ? V 1 V 2 ? D U 1 AV

?1

? U 1 AV
*

2

?0
* U 1 AV 2 ? ? D ? ? ? * U 2 AV 2 ? ? 0

?U 1* ? ?U 1* AV 1 * ? U AV ? ? ? A [V 1 , V 2 ] ? ? * * ?U 2 ? ?U 2 AV 1 ? ?

0? ? 0?

?D ? A ?U? ?0

0? ?V 0?

*

(SVD representation)
?D A ?U? ?0 0? ?V 0?
*

?D ? [u1 , ? , u m ]? ?0

? v 1* ? 0?? ? ?? ? ? ? 0? * ?v ? ? n?

??
i ?1

k

i

uiv ?
* i

?S
i ?1

k

i

18

Example 5.4.1. The singular value decomposition can be used for image compression. Consider all the singular values of A and order them greatest to least. Zero the matrix S for singular values less than some threshold.

19

Using 48 of 164 singular values

Using 16 of 164 singular values
20

CONCLUSION
Basic concepts:
? Hermitian matrix, skew-Hermitian matrix, positive define matrix, singular matrix. Important principles:

?
? ? ? ?

*eigenvalues of Hermiatian matrix are real.
*eigenvalues of PD matrix are positive. *diagonalizability of Hermitian matrix. * spectral representation for Hermitian matrix. * SVD.
21

HOMEWORK (2,3)
1. Section 4.1: (12)
?1 ? ?1 1? ? 1?

2. Proof

is PSD, not PD.

3. Proof the SVD theorem (in detail), and try to write down some applications of the SVD.
?5 ? ?3 3? ? 2?
1/ 2

4. Determine

.

22


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