Hessian and positive definiteness - what's the criteria?

So I read that if the determinant of a Hessian is positive and its the diagonal entries are both negative, then it can be said to be negative definite. However, what about the conditions for the others?
- Positive definiteness
- Positive semi-definiteness
- Negative semi-definiteness
Isn't there an equivalent sort of criteria?

Super confused and the test is coming up so would really appreciate any responses! Thank you x

Reply 1

mqb2766

21

Original post by olivier_

So I read that if the determinant of a Hessian is positive and its the diagonal entries are both negative, then it can be said to be negative definite. However, what about the conditions for the others?
- Positive definiteness
- Positive semi-definiteness
- Negative semi-definiteness
Isn't there an equivalent sort of criteria?

Super confused and the test is coming up so would really appreciate any responses! Thank you x

There are several equivalent definitions for things like positive definite
https://en.wikipedia.org/wiki/Definite_matrix
Assuming youre happy with eigenvalues, a simple way to test is
https://en.wikipedia.org/wiki/Definite_matrix#Eigenvalues

You can transform these to conditions like the one you mention. I take it youre referring to a 2*2?

Reply 2

olivier_

OP

12

Thank you for your reply! Yes it's a 2x2.
I haven't learned about eigenvalues yet - though I read the section from the link you sent me and it sounds very useful.
I will try to learn it, but somehow our teacher expects us to know how to classify the definiteness etc to identify local and global maxima/minima of f(x,y) without the knowledge of eigenvalues.
Do you know if there's a way to work it out with just the sign (+/-) of the determinant and the diagonal entries of the Hessian matrix?

Reply 3

mqb2766

21

Original post by olivier_

Thank you for your reply! Yes it's a 2x2.
I haven't learned about eigenvalues yet - though I read the section from the link you sent me and it sounds very useful.
I will try to learn it, but somehow our teacher expects us to know how to classify the definiteness etc to identify local and global maxima/minima of f(x,y) without the knowledge of eigenvalues.
Do you know if there's a way to work it out with just the sign (+/-) of the determinant and the diagonal entries of the Hessian matrix?

Yes, the simplest way is to think of just a diagonal matrix, so the diagonal entries are the eigenvalues.

The determinant is their product, the trace is their sum.

So if both (for a 2*2 symmetric? matrix) have the same sign, then the det is positive and the sign of the trace gives the negative or positive definite. If the determinant is zero, then at least one is zero and the sign of the trace gives the semi. pos/neg. Its basically the same as finding the (sign of) the roots of a "u" quadratic based on the sign of "b" (sum/trace) and "c".(product/determinant). Assuming the two roots are real, then they have the same sign if c is positive, and their sign corresponds to -sign(b) which is easy to understand by a quick sketch.

If the det is negative, they have opposite signs so correspond to a saddle. Again, thinking of a "u" quadratic with a negative "c" is insightful.

Obviously if you have a 3*3 or higher, then more conditions are needed and the above conditions are not necessarily valid.

(edited 1 year ago)

Hessian and positive definiteness - what's the criteria?

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