We know that the second derivative of a critical point is positive then we have a local minimum, if it is negative we have a local maximum. If we generalize this concept to functions with more variables, we must verify the sign of the Hessian Matrix H. It is said that if H is defined positive, ie:
$$x^THx>0$$
then we have a local minimum.
But why to say that the second derivative in a multidimensional space is positive, you must verify this condition?Why is x multiplied to the left and right of H? I might say something stupid, but wouldn't it be enough to verify that every component of the matrix H is positive and that's it? What is the real concept behind this definition?
Another definition, is that if H has all eigenvalues greater than zero, then it is defined positive, and then we have a local minimum. Why is this true? What is the concept behind this definition?
All this is true because H is a symmetrical matrix. Why is it important that H is symmetric, for the above definitions? If we don't have a symmetric matrix, how do we calculate its sign?