The Hessian Matrix is a square matrix of second ordered partial derivatives of a scalar function. It is of immense use in linear algebra as well as for determining points of local maxima or minima.
The above Hessian is of the the function where all second order partial derivatives of exist and are continuous throughout it's domain & the function is
The Hessian of a function is denoted by where is a twice differentiable function & if is one of it's stationary points then :
Usually Hessian in two variables are easy and interesting to look for. A function whose second order partial derivatives are well defined in it's domain so we can have the Hessian matrix of .
Note that the Hessian matrix here is always symmetric.
Let the function satisfies that it's second order partial derivatives exist & they're continuous throughout the Domain .
Then it's Hessian is given & denoted by :
The following example will demonstrate the facts clearly & explain it's uses.
Suppose a function is defined by . Find the maximum & minimum value of the function if it exists. Justify your answer.
We take the double derivatives of the function as follows :
The Hessian is defined by :
We solve for the Stationary points of the function by equating it's partial derivatives & to zero.
The possible pairings gives us the critical points .
Now as the Hessian consists of even functions which reduces a lot of effort. we only need to check for the pairs
For a brief knowledge of Definite & indefinite matrices study these first.
Now we check the Hessian at different stationary points as follows :
This is negative definite making it a local maximum of the function .
It's indefinite thus ruled out.
It's also indefinite
It's positive definite matrix and thus it's the local minimum of the function.
Thus we have successfully bounded the above function and it's point of local minimum is & point of local maximum is