Quadratic Form Example

General rule to compute arbitrary minors: 2 3 a11 a21 ::: an1 6 a12 a22 7 7 Take matrix A = 6 4 : 5 ::: a1n ann 2 a11 6 a22 Consider principle diagonal of matrix A : 6 4

3

::: ann

7 7 5

To compute arbitrary minors of order m (1 m n) : 1. Fix m elements of principle diagonal. 2. Delete all columns and all rows of matrix A that do not contain elements of principle diagonal that you have …xed. You are left with submatrix which have your chosen elements on principle diagonal. 3. Compute determinant of this submatrix. Note: there are

n! m!(n m)!

arbitrary minors of order m;where k! = 1 2 ::: k:

4. To show that matrix A positive(negative) semide…nite you have to check all arbitrary minors for each m: 1 m n: To show that matrix A is not positive(negative) semide…nite you have to …nd one arbitrary minor that does not satisfy condition m 0 (( 1)m m 0). Example 1 Consider quadratic form Q = x21 + x22 + 4x23 2x1 x2 4x1 x3 + 10x2 x3 1. Write down this quadratic form in matrix notation 30 2 1 x1 1 1 2 5 5 @ x2 A Q = x0 Ax = x1 x2 x3 4 1 1 2 5 4 x3 2.

Find all principal minors of matrix A: Is matrix A positive/negative

de…nite? D1 = 1

D2 =

1

1 1

1

1 2

1 5 1

1 D3 = ( 1)3+1 ( 2)

=1 1

1

1

( 1) ( 1) = 0

2 5 4 2 5

= ( 1)1+1 1 =

21 + 6 + 6 =

1

1 5

5 1 +( 1)2+1 ( 1) 4 5 9

2 4

+

D1 > 0; D2 = 0; D3 < 0 Matrix is neither positive de…nite nor negative de…nite 3. Find all arbitrary minors of matrix A: Is matrix A positive/negative semide…nite? m=1 Fix a11 : 1 =1>0 Fix a22 : 1 =1>0 Fix a33 : 1 =4>0 m=2 Fix a11 ; a22 :

2

=

Fix a11 ; a33 :

2

=

Fix a22 ; a33 :

2

=

1

1 1

1

2

4

1

1 5

2

5 4

=

=0

=0

21 < 0

m=3 1

1 2 1 1 5 = 9<0 2 5 4 Matrix is neither positive semide…nite nor negative semide…nite.

Fix a11 ; a22 ; a33 :

3

=

Quadratic form is not PD, ND, PSD, NSD, hence, quadratic form is indefinite Note: In this case to show that matrix is neither positive semide…nite nor negative semide…nite it is not necessary to compute all arbitrary minors. We know that 3 = det jAj = D3 : Before we found that D3 < 0: =)matrix A can not be positive semide…nite (necessary condition is m 0; 80 m n). As long as 1 > 0 matrix A can not be negative semide…nite (necessary condition is ( 1)m m 0; 80 m n; as long as m = 1 ( 1)1 1 = 0). 1

2