What is the definiteness of this function? I have this function:
$$\frac{1}{2}\left ( x^{2}+4y^2 \right )e^{-\left ( x^{2}+4y^2 \right )}$$
I might be wrong, but it is the product of two functions: the first one is positive definite (positive eigenvalues), while the exponential one is negative definite (negative eigenvalues). The answer tells me that this function is globally positive definite. But isn't the product of a positive definite function and a negative definite function a negative definite function? Where I'm wrong here?

In One ViewPoint, a function $P(x) , P(x,y) , P(x,y,z) ...$ is Positive Definite (or Positive Semi-Definite) when it is Positive (or Non-Negative) when evaluated at all values of $x , y , z ...$
In your case, :
$$P(x,y)= (x^2+4y^2)$$
$$f(x,y)= P(x,y)(e^{-P(x,y)})$$
Here $P(x,y)$ is 0 at (0,0) & Positive elsewhere; Hence it is Positive Semi-Definite.
The Power Part $-P(x,y)$ is 0 at (0,0) & Negative elsewhere; Hence it is Negative Semi-Definite.
Then $e^{-P(x,y)}$ is 1 at (0,0) & between 1 & 0 elsewhere; It can be considered Positive Definite.
End-Product is 0 at (0,0) & Positive elsewhere; Hence it can be considered Positive Semi-Definite.
