[math-fun] Positive eigenvalues imply positive definite?
I found this book review in the April 2020 Physics Today [ https://physicstoday.scitation.org/doi/10.1063/PT.3.4456 ]. The following is a quote from it. "After stating that “an arbitrary non-singular tensor T is positive definite if v·T·v>0 for all vectors v≠0,” the book goes on to spread the myth that positive eigenvalues of T are sufficient for T to be positive definite; the 2 × 2 matrix T={{4,9},{1,4}} with v={1,−1} is a counterexample." 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
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 In the forgetting of what has scarcely transpired there resonates the fury of one who must first talk himself out of what everyone knows, before he can then talk others out of it as well. — Theodor Adorno
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
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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How about the sequence of the number of n^2 x n^2 sudoku solutions that are positive definite? -Veit
On May 5, 2020, at 12:01 PM, Neil Sloane <njasloane@gmail.com> wrote:
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
participants (4)
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Cris Moore -
Eugene Salamin -
Neil Sloane -
Veit Elser