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test for definiteness of matrix

test for definiteness of matrix. Sylvester’s criterion and schur’s complement. outline. Why we test for definiteness of matrix? detiniteness . Sylvester’s criterion Schur’s complement conclusion. Why we test for definiteness of matrix?. Many application Correlation matrix

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test for definiteness of matrix

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  1. test for definiteness of matrix Sylvester’s criterion and schur’s complement

  2. outline • Whywe test for definiteness of matrix? • detiniteness. • Sylvester’s criterion • Schur’s complement • conclusion

  3. Whywe test for definiteness of matrix? • Many application • Correlation matrix • Factorization • Cholesky decomposition. • classification

  4. submatrix • k x k submatrix of an n x n matrix A deleting n − k rows and n − k columns of A • Principal submatrix of A deleted row indices and the deleted column indices are the same • leading Principal submatrix of A principal submatrix which is a north-west corner of the matrix A • Principal minor : determinant of principal submatrix • Leading principal minor : determinant of leading principal submatrix

  5. definiteness

  6. Positive definite matrix • Definition • A nxn real matrix M is positive definite if • Equivalence at real symmetric martixM • All eigenvalues of M > 0 • All leading principal minor > 0 • All diagonal entries of LDU decomposition > 0 • There exist nonsingular matrix Rs.t

  7. Negative definite matrix • Definition • A nxn real matrix M is negative definite if • Equivalence at real symmetric martixM • All eigenvalues of M < 0 • All leading principal minor of even size > 0 and all leading principal minor of odd size < 0 • All diagonal entries of LDU decomposition < 0

  8. Positive semi-definite matrix • Definition • A nxn real matrix M is positive semi-definite if • Equivalence at real symmetric martixM • All eigenvalues of M ≥ 0 • All principal minor ≥ 0 • All diagonal entries of LDU decomposition ≥ 0 • There exist possibly singular matrix Rs.t

  9. Negative semi-definite matrix • Definition • A nxn real matrix M is negative semi-definite if • Equivalence at real symmetric martixM • All eigenvalues of M ≤ 0 • All principal minor of even size ≥ 0 and all principal minor of odd size ≤ 0 • All diagonal entries of LDU decomposition ≤ 0

  10. Indefinite matrix • Definition • A nxn real matrix Mindetinite if and

  11. Sylvester’s criterion

  12. Positive definite matrix • Definition • A nxn real matrix M is positive definite if • Equivalence at real symmetric martixM • All eigenvalues of M > 0 • All leading principal minor > 0 • All diagonal entries of LDU decomposition > 0 • There exist nonsingular matrix Rs.t

  13. Negative definite matrix • Definition • A nxn real matrix M is negative definite if • Equivalence at real symmetric martixM • All eigenvalues of M < 0 • All leading principal minor of even size > 0 and all leading principal minor of odd size < 0 • All diagonal entries of LDU decomposition < 0

  14. Positive semi-definite matrix • Definition • A nxn real matrix M is positive semi-definite if • Equivalence at real symmetric martixM • All eigenvalues of M ≥ 0 • All principal minor ≥ 0 • All diagonal entries of LDU decomposition ≥ 0 • There exist possibly singular matrix Rs.t

  15. Negative semi-definite matrix • Definition • A nxn real matrix M is negative semi-definite if • Equivalence at real symmetric martixM • All eigenvalues of M ≤ 0 • All principal minor of even size ≥ 0 and all principal minor of odd size ≤ 0 • All diagonal entries of LDU decomposition ≤ 0

  16. Sylvester’s criterion • A nxn real symmetric matrix M is positive definite iff All leading principal minor > 0 • A nxn real symmetric matrix M is negative definiteiff All leading principal minor of even size > 0 and all leading principal minor of odd size < 0 • A nxn real symmetric matrix M is positive semi- definite iff All principal minor ≥ 0 • A nxn real symmetric matrix M is positive definite iff All principal minor of even size ≥ 0 and all principal minor of odd size ≤ 0

  17. Sylvester’s criterion • A nxn real symmetric matrix M is positive semi- definite iff all leading principal minor ≥ 0 : False! • /ex/ all leading principal minor ≥ 0 Exist 1 negative eigenvalue. It is not positive definite

  18. positive definite • A nxn real symmetric matrix M is positive definite iff All leading principal minor > 0 • sufficient condition real symmetric matrix M is positive definite ⇒let ⇒ ⇒kxk size leading principal minor

  19. positive definite • A nxn real symmetric matrix M is positive definite iff All leading principal minor > 0 • necessary condition kxk size leading principal minor ⇒kth diagonal entry of LDU decomposition ⇒ ⇒ real symmetric matrix M is positive definite

  20. negative definite • A nxn real symmetric matrix M is negative definite iff All leading principal minor of even size > 0 and all leading principal minor of odd size < 0 • sufficient condition real symmetric matrix M is negative definite ⇒let ⇒ ⇒kxk size leading principal minor if k is even if k is odd

  21. negative definite • A nxn real symmetric matrix M is negative definite iff All leading principal minor of even size > 0 and all leading principal minor of odd size < 0 • necessary condition ⇒i-th diagonal entry of LDU decomposition ⇒ ⇒ real symmetric matrix M is negative definite

  22. positive semi-definite • A nxn real symmetric matrix M is positive semi-definite iff All principal minor ≥ 0 • sufficient condition real symmetric matrix M is positive semi-definite ⇒le ⇒ is positive semi-definite ⇒ principal minor

  23. positive semi-definite • A nxn real symmetric matrix M is positive semi-definite iff All principal minor ≥ 0 • necessary condition principal minor ⇒let ⇒

  24. positive semi-definite • A nxn real symmetric matrix M is positive semi-definite iff All principal minor ≥ 0 • necessary condition

  25. positive semi-definite • A nxn real symmetric matrix M is positive semi-definite iff All principal minor ≥ 0 • necessary condition ⇒ ⇒ real symmetric matrix M is positive semi-definite

  26. negative semi-definite • A nxn real symmetric matrix M is negative semi-definite iff All principal minor of even size ≥ 0 and all principal minor of odd size ≤ 0 • sufficient condition real symmetric matrix M is negative semi-definite ⇒let ⇒ is negative semi-definite ⇒ principal minor

  27. negative semi-definite • A nxn real symmetric matrix M is negative semi-definite iff All principal minor of even size ≥ 0 and all principal minor of odd size ≤ 0 • necessary condition principal minor ⇒let ⇒

  28. negative semi-definite • A nxn real symmetric matrix M is negative semi-definite iff All principal minor of even size ≥ 0 and all principal minor of odd size ≤ 0 • necessary condition

  29. negative semi-definite • A nxn real symmetric matrix M is negative semi-definite iff All principal minor of even size ≥ 0 and all principal minor of odd size ≤ 0 • necessary condition ⇒ ⇒ ⇒ real symmetric matrix M is negative semi-definite

  30. Schur’s complement

  31. Schur’s complement • is positive definite iff and are both positive definite. • is positive definite iff and are both positive definite. • If is positive definite, is positive semi-definite iff is positive semi-definite • If is positive definite, is positive semi-definite iff is positive semi-definite

  32. Schur’s complement • is positive definite iff and are both positive definite. • sufficient condition is positive definite Let ⇒ and are both positive definite.

  33. Schur’s complement • is positive definite iff and are both positive definite. • necessary condition and are both positive definite. ⇒

  34. Schur’s complement • is positive definite iff and are both positive definite. • necessary condition ⇒ is positive definite ⇒ is positive definite

  35. Schur’s complement • is positive definite iff and are both positive definite. • sufficient condition is positive definite Let ⇒ and are both positive definite.

  36. Schur’s complement • is positive definite iff and are both positive definite. • necessary condition and are both positive definite. ⇒

  37. Schur’s complement • is positive definite iff and are both positive definite. • necessary condition ⇒ is positive definite ⇒ is positive definite

  38. Schur’s complement • If is positive definite, is positive semi-definite iff is positive semi-definite • sufficient condition is positive semi-definite Let ⇒ is positive semi-definite.

  39. Schur’s complement • If is positive definite, is positive semi-definite iff is positive semi-definite • necessary condition is positive definite. Is positive semi-definite ⇒

  40. Schur’s complement • If is positive definite, is positive semi-definite iff is positive semi-definite • necessary condition ⇒ is positive definite ⇒ is positive semi-definite

  41. Schur’s complement • If is positive definite, is positive semi-definite iff is positive semi-definite • sufficient condition is positive semi-definite Let ⇒ is positive semi-definite.

  42. Schur’s complement • If is positive definite, is positive semi-definite iff is positive semi-definite • necessary condition is positive definite. Is positive semi-definite ⇒

  43. Schur’s complement • If is positive definite, is positive semi-definite iff is positive semi-definite • necessary condition ⇒ is positive definite ⇒ is positive semi-definite

  44. Sylvester’s criterion • A nxn real symmetric matrix M is positive definite iff All leading principal minor > 0 • A nxn real symmetric matrix M is negative definiteiff All leading principal minor of even size > 0 and all leading principal minor of odd size < 0 • A nxn real symmetric matrix M is positive semi- definite iff All principal minor ≥ 0 • A nxn real symmetric matrix M is positive definite iff All principal minor of even size ≥ 0 and all principal minor of odd size ≤ 0

  45. Schur’s complement • is positive definite iff and are both positive definite. • is positive definite iff and are both positive definite. • If is positive definite, is positive semi-definite iff is positive semi-definite • If is positive definite, is positive semi-definite iff is positive semi-definite

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