So I have this math final coming up on Wednesday, and recently we have been finding critical points for two variable function using the Hessian matrix, and we didn't really explicitly learn how to find out the definiteness of the Hessian matrix in order to determine whether the point is a min/max etc.
I was wondering if there is a good way to be able to find this out, something quick and simple that I could apply in the exam.
So far in the couple of example we had, the prof. said that if the value of the top left element of the matrix is greater than 0, as well as the determinant, then it's positive semi-definite. But I think this only works for positive matrices.
We're more than likely only going to be dealing with real numbers, rather than complex numbers, so I'm looking for something applicable for real matrices.
Thank you!
Best Answer
I know these two criteria: A symmetric matrix is positive definite iff all eigenvalues are greater zero, positive semidefinite iff alle eigenvalues are greater or equal zero, and analogous for negative eigenvalues. It is indefinite, if there are positive and negative eigenvalues.
Then, there's Sylversters Criterion, which says that a symmetric matrix is positive definite iff the leading principial minors have positive determinant, which is in your case the top-left element and the 2x2-determinant. For positive semi-definiteness, you have to consider all principal minors. Check out https://en.wikipedia.org/wiki/Sylvester%27s_criterion.
Example: Let $A = \begin{pmatrix} 1 & 2 \\ 2 & -3 \end{pmatrix}$. Since $\det(1) = 1 > 0$ and $\det(A) = -7$, the matrix is not positive definite. But the characteristic polynomial is $\chi(x) = x^2 + 2x - 7$ and has a positive and a negative root, thus $A$ has a positive and a negative eigenvalue, so it is indefinite.