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We know that any matrix of the form $A^TA$ is positive semi-definite where $A^T$ is the transpose of $A$. Now how can we use this result in optimization?

Edit: The importance of positive semi-definite matrices is almost clear, but my question is specific to $A^TA$. I have no idea of how we can use this matrix.

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marked as duplicate by hxd1011, jbowman, mdewey, kjetil b halvorsen, AdamO Jan 30 '18 at 19:41

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  • $\begingroup$ @hxd1011: Thank you. Yes, the importance of SPD's is almost clear, but my question is specific to $A^TA$. I have no idea of how we can use this matrix. $\endgroup$ – M. Er Jan 30 '18 at 17:06
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    $\begingroup$ Can we have some more context on your particular problem? As is, it's excessively broad. A narrower version of your question (specialized to the scalar case) is "we know that $x^2$ is non-negative. How can we use this result in optimization?" How could someone answer that? You could fill a book with examples where a variable is squared. $\endgroup$ – Matthew Gunn Jan 30 '18 at 17:18
  • $\begingroup$ Any positive semi-definite matrix can be factored as $A^T A$ via a number of non-unique representations (Cholesky factorization, symmetric matrix square root, ...). So your question is seems to be asking about SPD matrices in optimization generally, which is pretty broad! $\endgroup$ – Andrew M Jan 30 '18 at 17:21