Why Matlab tells the following A*A^T matrix is not a positive Semi-definite Matrix ?

M = [ 1.0000      0         0         0         0         0;...
         0    0.9803   -0.0000   -0.0000   -0.0984    0.0984;...
         0   -0.0000    0.9902   -0.0984    0.0000    0.0000;...
         0   -0.0000   -0.0984    0.0098    0.0000   -0.0000;...
         0   -0.0984    0.0000    0.0000    0.0099   -0.0099;...
         0    0.0984    0.0000   -0.0000   -0.0099    0.0099];

Is from and its eigenvalues are

   d =

   -0.0000
   -0.0000
    0.0000
    1.0000
    1.0000
    1.0000  = 
    
  %When vpa is used it shows 
   -7.365e-18
    -2.12e-18
    1.347e-16
        1.0
        1.0
        1.0
    

So, can't we call matrix M, positive semidefinite ?

Apperciated!

Answer 1

[Matt J]

Yes, it is positive semi-definite. But Matlab's ability to detect that is limited, because finite precision prevents it from computing exact eigenvalues.

[HN]

Thank you Matt J ,

But how can I convince someone ?Any way to proof this fact ?

[Matt J]

Well, it is a known result from linear algebra that A*A^T is always positive semidefinite. There must be a million text books you could reference.

[HN]

Ok, I read some theorem that states A*A^T could be negative or positive semidefinite. That is why.

Thank you

[Matt J]

It is very easy to prove from the definition of positive semidefiniteness

x.'*(A*A.')*x
=(x.'*A)*(A.'*x)
=(A.'*x).' * (A.'*x)
=dot(A.'*x,A.'*x)
>=0

[HN]

Thank you Matt J ,