## Hessian Matrix

### Introduction

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(x_{1}, x_{2}, ..., x_{n}), 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 = (x_{1}, x_{2}, ..., x_{n}) and D_{i} is the differentiation operator with respect to the *i*th argument. Thus

**Defination from an existing Jacobian matrix**

The Jacobian matrix of the derivatives , , ..., of a function with respect to x_{1}, x_{2}, ..., x_{n} is called the Hessian ** H** of f, i.e.

### Applications

Hessian matrices are used in large-scale optimisation problems within Newton-type methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is

### Vector-valued functions

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.

### References & Resources

- http://en.wikipedia.org/wiki/Hessian_matrix
- http://mathworld.wolfram.com/Hessian.html

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