test for definiteness of matrix

TEST FOR DEFINITENESS OF MA-TRIX

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

Factorization Cholesky decomposition.

classification

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 Aprincipal submatrix which is a north-west corner of the ma-

trix A Principal minor : determinant of principal submatrix Leading principal minor : determinant of leading prin-

cipal submatrix

definiteness

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

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

Positive semi-definite matrix

Definition A nxn real matrix M is positive semi-defi-

nite 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

Negative semi-definite matrix

Definition A nxn real matrix M is negative semi-defi-

nite 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

Indefinite matrix Definition

A nxn real matrix M indetinite if and

Sylvester’s criterion

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

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

Positive semi-definite matrix

Definition A nxn real matrix M is positive semi-defi-

nite 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

Negative semi-definite matrix

Definition A nxn real matrix M is negative semi-defi-

nite 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

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

Sylvester’s criterion A nxn real symmetric matrix M is positive

semi- definite iff all leading principal mi-nor ≥ 0 : False!

/ex/

all leading princi-pal minor ≥ 0

Exist 1 negative eigenvalue.It is not positive definite

positive definite A nxn real symmetric matrix M is positive

definite iff All leading principal minor > 0

sufficient conditionreal symmetric matrix M is positive definite⇒ let ⇒

⇒ kxk size leading principal minor

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

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 conditionreal symmetric matrix M is negative definite⇒ let ⇒

⇒ kxk size leading principal minor if k is even if k is odd

negative definite A nxn real symmetric matrix M is negative defi-

nite 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

positive semi-definite A nxn real symmetric matrix M is positive

semi-definite iff All principal minor ≥ 0

sufficient conditionreal symmetric matrix M is positive semi-definite⇒ le

⇒

is posi-tive semi-definite

⇒ principal minor

positive semi-definite A nxn real symmetric matrix M is positive

semi-definite iff All principal minor ≥ 0

necessary conditionprincipal minor

⇒ let

⇒

positive semi-definite A nxn real symmetric matrix M is positive

semi-definite iff All principal minor ≥ 0

necessary condition

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

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 conditionreal symmetric matrix M is negative semi-definite⇒ let⇒

is nega-tive semi-definite

⇒ principal minor

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 conditionprincipal minor

⇒ let⇒

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

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

Schur’s complement

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-defi-nite iff is positive semi-definite

If is positive definite, is positive semi-defi-nite iff is positive semi-definite

Schur’s complement

is positive definite iff and are both positive definite.

sufficient condition is positive definite Let

⇒ and are both positive definite.

Schur’s complement

is positive definite iff and are both positive definite.

necessary condition and are both positive definite.

⇒

Schur’s complement

is positive definite iff and are both positive definite.

necessary condition ⇒

is positive definite

⇒ is positive definite

Schur’s complement

is positive definite iff and are both positive definite.

sufficient condition is positive definite Let

⇒ and are both positive definite.

Schur’s complement

is positive definite iff and are both positive definite.

necessary condition and are both positive definite.

⇒

Schur’s complement

is positive definite iff and are both positive definite.

necessary condition ⇒

is positive definite

⇒ is positive definite

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.

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

⇒

Schur’s complement

If is positive definite, is positive semi-definite iff is positive semi-definite

necessary condition ⇒

is pos-

itive definite

⇒ is positive semi-definite

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.

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

⇒

Schur’s complement

If is positive definite, is positive semi-definite iff is positive semi-definite

necessary condition ⇒

is pos-

itive definite

⇒ is positive semi-definite

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

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-defi-nite iff is positive semi-definite

If is positive definite, is positive semi-defi-nite iff is positive semi-definite