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One of my homework solutions stated the following:

enter image description here

However, I do not seem to grasp why the solutions refers to 'both principal minors'. As far as I know there are three principal minors:

Order 1: det (2) = 2 and det (4) = 4

Order 2: det (2 12 ; 12 4) = -136

Wouldn't this mean that the Hessian matrix is NOT positive definite as the principal minor of order 2 is non-positive?

It might be important to show what f was defined as:

enter image description here

So my question now is whether the conclusion of the solution is correct: is the principal minor of order 2 not relevant to establish positive semi-definiteness?

Thank you very much in advance.

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  • $\begingroup$ Thank you very much for your input. First of all, would the leading principal minors then be 2 and 0? Furthermore, could you please elaborate on the Hessian matrix? The off-diagonal elements are the result of taking the partial derivative of x1 followed by the partial derivative of x2 (order is not relevant), right? $\endgroup$ – MathNoob123 Mar 10 at 22:03
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    $\begingroup$ Positive definiteness requires the $2$ leading principal minors to be positive. You computed both of them correctly. You are also correct that there are $2^2 - 1 = 3$ principal minors; if all $3$ of them are non-negative, one can conclude positive semidefiniteness. $\endgroup$ – Rodrigo de Azevedo Mar 10 at 22:08

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