I was explained that in order to determine if a point is a saddle point I I should check if the determinant of the hassian matrix is negative.
Then I was looking online and found an article saying that we can just check if we have one positive Eigenvalue and one negative (and all other Eigenvalue are not zero)
If these 2 conditions are actually equivalent, I would really appreciate to get some intuition for why they are equivalent
For functions from $\Bbb R^2$ into $\Bbb R$, it is true. Asserting that the determinant of the Hessian is negative is the same thing as asserting the there are eigenvalues with different signs.
But it is false for functions from $\Bbb R^3$ into $\Bbb R$. Take $f(x,y,z)=-x^2-y^2-z^2$ at $(0,0,0)$. The determinant of the Hessian there is $-8$, but $f$ has a maximum at that point: all eigenvalues are negative.