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

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HN
HN on 22 Oct 2020
Commented: HN on 22 Oct 2020
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!

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Accepted Answer

Matt J
Matt J on 22 Oct 2020
Edited: Matt J on 22 Oct 2020
Yes, it is positive semi-definite. But Matlab's ability to detect that is limited, because finite precision prevents it from computing exact eigenvalues.

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HN
HN on 22 Oct 2020
Ok, I read some theorem that states A*A^T could be negative or positive semidefinite. That is why.
Thank you
Matt J
Matt J on 22 Oct 2020
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

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