This point came up a few years ago (around 2003) in connection with counting n X n real {0,1} matrices which are (i) positive definite, or (ii) have all eigenvalues positive, etc. See A003024, A085656, plus there are several related sequences. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sun, May 3, 2020 at 2:42 PM Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
Yes, if A is symmetric, then A can be diagonalized with an orthogonal similarity matrix S. Then
x^T A x = x^T (S^T D S) x = (Sx)^T D (Sx) > 0.
-- Gene
On Sunday, May 3, 2020, 11:22:57 AM PDT, Cris Moore via math-fun < math-fun@mailman.xmission.com> wrote:
this holds if A is symmetric, right? the counterexample is not.
On May 3, 2020, at 12:16 PM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
Then in reading Arthur Gelb, "Applied Optimal Estimation", I found problem 2-3, in which A is a matrix.
"Show that A is positive definite if and only if all of its eigenvalues are positive."
-- Gene
Cris Moore moore@santafe.edu
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