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Inverse of a positive semi-definite matrix?

by mikeph
Tags: inverse, matrix, positive, semidefinite
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Dec5-12, 08:49 AM
P: 1,212
If A is some nonsquare matrix that is possible rank-deficient, 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
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Dec5-12, 09:16 AM
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Any real nxm matrix A will have ATA (and AAT) be positive semidefinite.

Now let A be some matrix all of whose elements are zero. Obviously both ATA and AAT 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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