Download 2.4 A square-root matrix

1
2.4 A Square-Root Matrix:
Suppose the matrix A is a positive definite matrix. Thus,
A  PDP t  ( PD1/ 2 Pt )( PD1/ 2 Pt )  A1/ 2 A1/ 2 ,
where D1 / 2 is also a diagonal matrix with diagonal elements equal to
the square root of those diagonal elements of D. Therefore,
A1/ 2  PD1/ 2 P t ,
is called the square root of A.
Properties of A1 / 2 :
1. A
1
n

2
i 1
i wi wit
t
1
1
A 2
  A 2 (that is, A1 / 2 is symmetric).
2. 


1
3.
2
A A
1
1

2 
A

4. 


 A
1
2
 PD
1
2
P 
t
n
1

wi wit ,
i
i 1
i .
1
5.
1
D 1 / 2 is a diagonal matrix with diagonal elements equal to
where
1
 A.
2
2
A A
1
2
A
1
2
A
1
2
I
and
A
1
2
2.5 Random Vectors and Matrices:
Let
X ij , i  1,, n, j  1,, p,
be random variables. Let
 X 11 X 12
X
X 22
21

X
 


 X n1 X n 2
 X1p 
 X 2 p 
  

 X np 
A
1
2
 A 1 .
2
be the random matrix.
Definition:
 E ( X 11 ) E ( X 12 )
E( X ) E( X )
21
22
EX   
 


 E ( X n1 ) E ( X n 2 )
 E( X1p ) 
 E ( X 2 p )
 E X ij  n p

 
.

 E ( X np ) 


Theorem:
 
 
Aln  aij , B pk  bij are two matrices, then
E AXB  AE X B .
Results:



 
 E X n p  Z n p  E X n p  E Z n p

 E  Amn X n1  BmnYn1   AE X n1   BE Yn1 
Example 11:
Consider two random variables X 1 and X 2 with probability functions
0.2, x1  1
0.5, x2  0

,
f1 x1    0.6, x1  0 and f 2 x2   
 0.5, x2  1
 0.2, x  1
1

respectively. Then,
E X 1    1  0.2  0  0.6  1 0.2  0
and
E X 2   0  0.5  1 0.5  0.5 .
Thus, for X  X 1 , X 2  ,
t
 E  X 1   0 
E X   
  0.5


E
X
 
2 
