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Hessian matrix positive/negative definite watch

1. Hi,

Having trouble wrapping my head around working out whether a hessian matrix is positive, positive semi, negative or negative semi definite.

E.g. a 2x2 matrix:

a11 a12
a21 a22

am I right in thinking it

is positive definite if a11 > 0 and the determinant > 0 (minimum)
is negative definite if a11 < 0 and determinant is > 0 (maximum)
indefinite if the determinant is < 0 (saddle point)

Have no idea how to check for semi definite though, if anyone could help or correct me it would be much appreciated
2. (Original post by Scorcher)
Hi,

Having trouble wrapping my head around working out whether a hessian matrix is positive, positive semi, negative or negative semi definite.

E.g. a 2x2 matrix:

a11 a12
a21 a22

am I right in thinking it

is positive definite if a11 > 0 and the determinant > 0 (minimum)
is negative definite if a11 < 0 and determinant is > 0 (maximum)
indefinite if the determinant is < 0 (saddle point)

Have no idea how to check for semi definite though, if anyone could help or correct me it would be much appreciated
No. The matrix {{5, 17}, {0, 11}} is indefinite: gives , but det M is 55.

Possibly the quickest way to work out these things: a real matrix is positive/negative definite iff the symmetric matrix is pos/neg def. A symmetric matrix is pos/neg def iff all its eigenvalues are positive/negative.

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Updated: December 24, 2014
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