I am trying to deal with an issue: I am trying to determine the nature of some points, that's why I need to check in Matlab if a matrix with complex elements is positive or negative definite. After performing some research, I came to the following two methods:


Calculate the eigenvalues and see if it is positive/negative;

    eig(matrix)


If the eigenvalues are positive => matrix is positive definite;
Else, if eigenvalues are negative => matrix is negative definite;
Use the following function:

    [R P] = chol(matrix)


If p is 0 => you have a positive definite matrix; otherwise, your matrix is NOT positive definite.


My problem is that I have two complex eigenvalues (and  my symmetric matrix has complex elements), therefore, method 1 doesn't help me to draw any conclusion. And the method 2, doesn't give me information whether the matrix is negative definite or indefinite, because it tests only if the matrix is positive definite or not, therefore, doesn't solve my problem.

Does any one have any idea how I can check if a matrix with complex eigenvalues is positive or negative definite with other methods than the mentioned ones? Thank you.

LATER EDIT:

My function is: 11.*x1 + 22.*x1.^2.*x2 + x2.^2 + 31.*x1.^2;

I have done the partial derivative with respect to x1, x2 and I have equalized all the obtained relations with 0. Therefore, I have obtained the following system:

62*x1 + 44*x1*x2 + 11 = 0

22*x1^2 + 2*x2 = 0

from here, I got x1 having the following possible values: 
{0.333, -0.1025 + 0.2403i, -0.1025 - 0.2403i} 
and x2 having the following values: 
{-1.22, 0.5197 + 0.5417i,  0.5197 - 0.5417i}

My Hessian looks something like:

( 62+44*x2                 44*x1)

( 44*x1                    2    )

After taking the 3 possible pairs of (x1,x2) I obtain three values for the Hessian matrix. For the first (x1,x2) pair everything is ok => I have a saddle, since I have 2 real eigenvalues of opposite signs. The difficulties comes with the others, because intuitively I would say that we cannot determine the nature of those points, but I came across to the following idea, while surfing the internet for finding an explanation:

-if A belongs to Mn(C) and 

1) Re(x_star * A * x) > 0 => A positive definite.

2) Re(x_star * A * x) &lt; 0 => A negative definite.

(according to http://mathworld.wolfram.com/PositiveDefiniteMatrix.html)

So, I am a bit confused :-?


Question: If $a$ is a quadratic residue of odd prime $p$, then is it the case that $a^{-1}$ is also a quadratic residue?
Answer: Maps like $\,x\mapsto x^{-1}$ that are multiplicative $\,h(ab) = h(a)h(b)$ preserve squares $\,h(a^2)= (h(a))^2$