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Corollary 1.2.9
If A is diagonalizable and rank A=r,
then A has at least one rxr nonsingular
principal submatrix.
permutation
 S : symmetricmatrix
group
n
P  e (1)
e ( 2 )  e ( n ) : permutation matrix
 A1* 
A 
2*


Let A 
  
 
 An* 
AP  A* (1)
 A (1)* 
A 
T
 ( 1 )*


P A 
  


 A (1)* 
A* ( 2 )  A* ( n ) 
Proof
 A (1)* 
A 
T
 ( 2 )*


P A 
  


 A ( n )* 
 A1* 
A 
T
T
2*

  A ( i )*
( P A)i*  ( P )i* A  e ( i )
  
 
 An* 

Proof AP  A* (1)
A* ( 2 )  A* ( n ) 
( AP )* j  A( P )* j  A*1 A*2  A*n e ( j )  A* ( j )
( P AP )ij  A ( i ) ( j )
T
Proof
( P AP )ij  e ( i ) Ae ( j )  e ( i ) A* ( j )  A ( i ) ( j )
T
T
T
Fact 1.2.10
p.1
If A is mxn matrix and r is the size of
the largest nonsingular submatrix.Then
(i)
rank A=r
(ii)
If B is a rxr nonsingular submatrix,
then there are permutation matrices
Fact 1.2.10
p.2
P,Q such that
I 
P AQ    BI
 R
T
S
(iii) If , in addition, m=n and B is principal then
may choose
QP
Let B  b1
b2
 br rxr
be a non sin gular submatrix of
Proofn of
Fact 1.2.10
 permutatio
matrices
P, Q s.t
 B *
P AQ  

 * *
Let P T AQ  a1
A
p.1
T
Since b1 ,
a1 ,
a2
 an mxn
b2 ,  , br are linear independen t
a2 ,  , ar are linear independen t
To establish (i ), sufficies to show that
each a j , r  1  j  n, is a linear
combination of
a1 ,
a2 ,  , ar
a  

j
Let a j   
where a j is r  1 matrix
 * 
Proof
of
Fact
1.2.10
p.2

then a j  s1 j b1  s2 j b2    srjbr
Claim : a j  s1 j a1  s2 j a2    srjar
pf : If not ,  r  1  k  m s.t
the correspond ing entries are not equal .
B a 
j
Then 
is ( r  1)  ( r  1) non sin gular. 

 * *  ( r 1)( r 1)
kth row
B
Let P AQ  
C
D
 a1 a2  an 

E
Proof of Fact 1.2.10
p.3
Since each a j , r  1  j  n, is
T
a linear combination of a1 , a2 ,  , ar ,
 D  B
 BS 
for some S  M r ( n  r )
 E   C  S  CS 
   
 
 B D   B BS   B 
 

  I S 


C E  C CS  C 
 1 
 
B
   2
Let   
Proof
  Fact 1.2.10
C   of
 
 m 
Since each  j , r  1  j  m, are
p.4
a linear combination of 1 ,  2 ,  ,  r ,
C  RB
for some R  M ( m  r )r
B
 
C 
 B  I 
 RB    R  B
   
B
I 
T
 P AQ    I S    B I
C 
 R
S
If B is principal , then
Proof of Fact 1.2.10
p.5
we may choose P  Q s.t
B
*


T
Q AQ  

 * *
Theorem 1.2.13
A  M m n , B  M n m , m  n
 cBA (t )  t cAB (t )
n m
If m=n ,and at least one of A or B is
nonsigular,then
AB and BA are similar
 AB 0  I m A   AB ABA
(1) 





0  0 I n   B
BA 
 B
Proof of Theorem 1.2.13
p.1
0   AB ABA
 I m A  0
 0 I   B BA    B

BA 
n 
 

0 
 AB 0  I m A   I m A   0








0  0 I n   0 I n   B BA 
 B
1
 I m A   AB



 0 In   B
 AB 0  0

~

0  B
 B
0  I m


0  0
0 

BA 
A  0


In  B
0 

BA 
then they have the same chara . poly .
 cAB Proof
(t )t  of
t cTheorem
(t )
BA
1.2.13
nm
 cBA (t )  t cAB (t )
n
m
p.2
( 2) If m  n and A is non sin gular
then A ( AB) A  BA
1
 AB and BA are similar .
Corollary 1.4.3
If A is a real symmetric matrix of rank r
then there is a permutation P and
rxr nonsingular principal submatrix M s.t
I 
P AP    M I
 R
T
R
T

for some matrix R
Since A is real symmertic ,
A is diagonaliz able
of Corollary
1.4.3
 by Proof
Corollary
1.2.9
p.1
A has a principal submatrix M of order r
By Corollary 1.2.10
 permutation matrix P s.t
I 
P AP    M I S 
 R
I 
M
   M MS   
 R
 RM
T
MS 

RMS 
T
But P AP is symmertic ,
MS
M

Proof of Corollary
1.4.3
hence so is 

 RM RMS 
T
T
T
T
 RM  ( MS )  S M  S M
RS ,
T
sin ce M is non sin gular
I 
 P AP    M I
 R
T
p.2
R
T

 x1   y1 
Usual
 Inner
 Product of C
n
x
y
 2  2
,

x
y

r
r
     
r 1
   
x
y
 n  n
*
 y x
n
i.e
x, y  y x
*
Unitary
U is said to be
and equals
i.e
unitary if U
U
*
U U  UU  I
*
*
1
exists
Fact
U  Mn
is
unitary if and only if
the columns of U form an
orthonormal basis of
proof: (see next page)
C
n
u1 
u * 
2
*

U U  I  u1 u2  un   I n

 *
 un 
*
 ui u j   ij
*
 u j , ui   ij
 i, j
 i, j
 u1 , u2 ,  , un  is an
orthonormal basis of C
n
Real Orthogonal
Q  M n (R)
is real orthogonal if
QQ  Q Q  I
T
T
i.e A real orthogonal matrix is a real
matrix which is unitary
Fact
U  M n (R) is real orthogonormal if and
only if the columns of U form an
orthonormal basis of
proof: (see next page)
R
n
u1 
u T 
2
T

u1 u2  un   I n
UUI

 T
 un 
T
 ui u j   ij
T
 u j , ui   ij
 i, j
 i, j
 u1 , u2 ,  , un  is an
orthonormal basis of R
n
Fact
A is unitarily diagonalized
C
n
has a orthonormal basis
consisting of eigenvectors of A
proof: (in next page)
0
1


2
*

U AU  





n 
0
0
1


2

 Au1 u2  un   u1 u2  un 





n 
0
 Au j   j u j
 j  1,2,, n
 u1 , u2 ,  , un  form a orthogonomal basis
of C
n
consisting of eigenvectors of
A
Fact
A is diagonalizable
C
n
has a basis consisting of
eigenvectors of A
proof: (in next page)
0
1


2
1

P AP  





n 
0
0
1

2
 A p1 p2  pn    p1 p2  pn 




0
 Ap j   j p j
 j  1,2,, n
 p1 ,
p2 ,  , pn  form a basis of C n
consisting of eigenvectors of
A
Theorem 1.4.1(The spectral Thm for
Hermitian matrices)
p.1
A is Hermitian   unitary matrix U s.t
1

*
U AU  


0

2
0





n 
?(未證)
where 1 , 2 ,, n are real
Theorem 1.4.1(The spectral Thm for
Hermitian matrices)
If A is real sysmmetric, then
U can be chosen to be
real orthogonal matrix
p.2