Suppose matrix $A$ has rank $k$. If all the eigenvalues of $A^TA$ are real, then is it true that $A^TA$ has $k$ positive eigenvalues(including repeated ones)? Can this be addressed using rank nullity theorem? Appreciate if one could advise me on that. Thank you.
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$A^TA$ is symmetric. If all of its entries are real, then so are all of its eigenvalues. – amd Sep 20 '19 at 05:26
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Does it mean that $k$ of the eigenvalues are positive? – Alexy Vincenzo Sep 20 '19 at 05:27