The Hessian Matrix is a square matrix of second-order partial derivatives of a function. It describes the local curvature of a function of many variables.
Defination from a function
Given the real-valued function f(x1, x2, ..., xn), if all second partial derivatives of f exist and are continuous over the domain of the function, then the Hessian matrix of f is
where x = (x1, x2, ..., xn) and Di is the differentiation operator with respect to the ith argument. Thus
Defination from an existing Jacobian matrix
The Jacobian matrix of the derivatives , , ..., of a function with respect to x1, x2, ..., xn is called the Hessian H of f, i.e.
If f is instead a function from ,i.e.
then the array of second partial derivatives is not a two-dimensional matrix of size , but rather a tensor of order 3. This can be thought of as a multi-dimensional array with dimensions , which degenerates to usual Hessian matrix for m=1.
Note: The Vector-valued function can be found from Jacobian matrix.
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