I read somewhere that in order to have a hessian matrix and be able to conduct further investigation to it, all the variables must be differentiable and they must all exist. Now i have come up with a hessian matrix in which i have 3 zeros. One in the diagonal and two in the off-diagonal part. I want to make sure that this matrix satisfies the above-mentined criterion and that i can continue with the manipulation of the matrix. Thanks in advance
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Yes, Hessians can have zeros in them. Consider any function which is a linear combination of functions solely in one variable (for example, $x^3 + y^3 + z^3$), these will always have zeros in any cross-terms. – Osama Ghani Nov 15 '22 at 21:49
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Sure, take $f(x,y)= x^2+y^2$. – copper.hat Nov 15 '22 at 21:56