Sylvester's Law of Inertia

[mihir_naik]

Dear all,

 

I have a query regarding the Sylvester's Law of Inertia (Congruence Transformation). It says the following (taken from Gilbert Strang---- 4th edition, Page 324) :

 

C^T*A*C has the same number of positive, negative and zero eigenvalues as A, for some nonsingular matrix C.

 

That is, the signs of eigenvalues are preserved by the congruence transformation.

 

Here, I give 2 matrices A and C^T*A*C where the signs of the eigenvalues do not seem to be preserved:

 

A =
-10.0000 -0.2000 -0.1500 5.0000 0.1440 0.3600
5.0000 -2.0000 0 -2.5000 1.4400 0
0 1.0000 -3.0000 0 -0.7200 7.2000
-5.0000 0 0 2.5000 0 0
0 -1.0000 0 0 0.7200 0
0 0 -1.5000 0 0 3.6000

 

Eigenvalues of A = -7.3508, -1.3815, 0, 0, 0, 0.5524 (Note that A is a singular matrix and hence it has zero eigenvalues)

 

C = 0.3162 0 0 0 0 0
0 0.7071 0 0 0 0
0 0 0.5774 0 0 0
0 0 0 0.4472 0 0
0 0 0 0 0.8333 0
0 0 0 0 0 0.3727

 

C^T*A*C =

 

-0.66332, -0.0000094371,0, 0.0002162, -0.4184-0.2645*i, -0.4184 + 0.2645*i

 

It is to be noted that C^T*A*C has two complex conjugate eigenvalues (and because A is singular, C^TAC is also singular), whereas A has only real eigenvalues. But it can be seen that A has two negative eigenvalues whereas C^T*A*C seems to have 4 negative eigenvalues (including the complex conjugates).

 

Any help on this issue is appreciated. Thanks.

[uncool]

You may want to check if the theorem is supposed to apply to symmetric matrices; that is where this kind of congruence transformation usually happens (otherwise, usually C^(-1) A C is used).

=Uncool-

[studiot]

My windows2000 mathcad gets the same eignvalues for A as you do, but different ones for the transformed matrix.

 

I get the three zeros again, your negative -0.66 and the complex conjugate pair.

 

I think it was Euler had a method to make the complex pair equivalent to a two real values.

Edited July 7, 2013 by studiot