Inverse of a positive semi-definite matrix?

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SUMMARY

The discussion confirms that for any real matrix A of size n x m, the product (A^T)(A) results in a positive semidefinite matrix. However, if A is rank-deficient or consists entirely of zeros, the resulting matrix (A^T)(A) becomes a zero matrix, which does not possess an inverse. The distinction between positive definite and positive semidefinite matrices is emphasized, highlighting the implications for matrix invertibility.

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mikeph
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Hi,
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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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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