Currently in a class dealing with this type of information currently, my question is an extension of the post: Principal Minor criteria to determine the nature of critical points

The user posted now when is the Hessian positive definite, negative definite, and indefinite? However, from the principal minor criteria, can't we not tell when the matrix/Hessian is negative definite or indefinite, just positive definite?

In class we had the definition for the principle minor criteria: "Let $A$ be an $n$x$n$ matrix, symmetric. Then $A$ is positive definitive if and only if $D_{1}(A)>0$, $D_{2}(A)>0$, ... , $D_{n}(A)>0$."

If my teacher asked me to use the principal minor criteria to find what we can determine about the nature of the critical points, would all be able to say is "Well I know A isn't positive definitive" (if there was a negative determinant of A) or am I missing something from the theorem? I know I'm able to tell the Hessian is positive/negative definite or indefinite from other theorems such as the sign of diagonal matrices, however from looking on material online I'm wondering if something is missing from the definition, or I'm understanding it wrong.

Thanks for reading my post, apologies if formatting is off.

  • 1
    $\begingroup$ If the leading principal minors of the matrix are alternatively positive and negative, it is negative definite. If some of them are positive, some negative or if some are positive,negative and zero, then the matrix would be indefinite. $\endgroup$ – StubbornAtom Nov 16 '17 at 4:17
  • $\begingroup$ Where is this in definition? I look online and can't find it at all. Also, how would you be able to tell whether or not it is semi-definite? @StubbornAtom $\endgroup$ – Riley Carney Nov 16 '17 at 4:19
  • $\begingroup$ Your definition includes the criteria for positive definite only. $\endgroup$ – StubbornAtom Nov 16 '17 at 4:21
  • $\begingroup$ what I've found online does as well, i.e. en.wikipedia.org/wiki/Sylvester%27s_criterion is there a paper or link you can provide with more information for your definition? $\endgroup$ – Riley Carney Nov 16 '17 at 4:25
  • $\begingroup$ Search this site for the relevant posts on positive semi-definite criterion. $\endgroup$ – StubbornAtom Nov 16 '17 at 4:25

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