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Inverse of a positive semidefinite matrix? 
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#1
Dec512, 08:49 AM

P: 1,212

Hi,
If A is some nonsquare matrix that is possible rankdeficient, then am I right to understand that (A^T)(A) is a positive semidefinite matrix? Does there exist an inverse (A^T A)^1? Thanks for any help 


#2
Dec512, 09:16 AM

Mentor
P: 15,170

Any real nxm matrix A will have A^{T}A (and AA^{T}) be positive semidefinite.
Now let A be some matrix all of whose elements are zero. Obviously both A^{T}A and AA^{T} will also be zero matrices (but now square), and obviously, no inverse. There's a world of difference between positive definite and positive semidefinite. 


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