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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 - PowerPoint PPT Presentation
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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
