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

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

3. Why we 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 martix M
All eigenvalues of M > 0
All leading principal minor > 0
All diagonal entries of LDU decomposition > 0
There exist nonsingular matrix R s.t

7. Negative definite matrix
Definition
A nxn real matrix M is negative definite if
Equivalence at real symmetric martix M
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 martix M
All eigenvalues of M ≥ 0
All principal minor ≥ 0
All diagonal entries of LDU decomposition ≥ 0
There exist possibly singular matrix R s.t

9. Negative semi-definite matrix
Definition
A nxn real matrix M is negative semi-definite if
Equivalence at real symmetric martix M
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 M indetinite if
and

11. Sylvester’s criterion

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

13. Negative definite matrix
Definition
A nxn real matrix M is negative definite if
Equivalence at real symmetric martix M
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 martix M
All eigenvalues of M ≥ 0
All principal minor ≥ 0
All diagonal entries of LDU decomposition ≥ 0
There exist possibly singular matrix R s.t

15. Negative semi-definite matrix
Definition
A nxn real matrix M is negative semi-definite if
Equivalence at real symmetric martix M
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 definite iff 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.
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 definite iff 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.
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