Hessian Matrix Calculator

Hessian Matrix Calculator

Hessian matrix calculator finds the hessian matrix of 2 variables as well as 3 variables. This hessian calculator also evaluates the determinant of the hessian matrix.

What is the hessian matrix?

A hessian matrix is a square matrix that contains the second-order partial derivative of the function. The determinate of the hessian matrix at a given point informs us of the trend of the function.

Formula:

The formula of the hessian matrix of order 2x2 and 3x3 is given below.

For 3x3

$$H_g=det\begin{pmatrix}f_{xx}&f_{xy}&f_{xz}\\ \:f_{yx}&f_{yy}&f_{yz}\\ \:f_{zx}&f_{zy}&f_{zz}\end{pmatrix}$$

For 2x2

$$H_g=det\begin{pmatrix}f_{xx}&f_{xy}\\ f_{yx}&f_{yy}\end{pmatrix}$$

Examples of Hessian Matrix

Let’s discuss a numerical example in detail.

Example 

Find the value of the hessian matrix at different points (0,0,0), (1,0,0), (-1,-1,-1), and (1,1,1) of a function

(2yx⁴, 2y³, 2xz³).

Solution:

Step 1: We have a function that is 3 dimensional.

f(x,y,z) = 2yx⁴ + 2y³ + 2xz³

Step 2: General formula 

$$H_g=det\begin{pmatrix}f_{xx}&f_{xy}&f_{xz}\\ f_{yx}&f_{yy}&f_{yz}\\ f_{zx}&f_{zy}&f_{zz}\end{pmatrix}$$

$$=f_{xx}\left(f_{yy\cdot f_{zz}-f^2_{yz}}\right)-f_{xy}\left(f_{xy}\cdot f_{zz}-f_{yz}\cdot f_{xz}\right)+f_{xz}\left(f\left(xy\right)\cdot f_{yz}-f_{yy}\cdot f_{xz}\right)$$

Step 3: Find the partial derivatives

f_xx = 24x²y
f_xy = 8x³
f_xz = 6z²
f_xy = 8 x^3
f_yy = 12y
f_yz = 0
f_xz = 6z²
f_yz = 0
f_zz =12xz

Step 4: Put the values in the formula and obtain the Hessian matrix.

$$H_g=\begin{pmatrix}2x^2y&8x^3&6z^2\\ \:8x^3&12y&0\\ \:6z^2&0&12xz\end{pmatrix}$$

Step 5: Now Find the determinate

= 192x⁵y – 48x³z²

Step 6: To check the value at the given point, we have

Dg _(0,0,0) = 0

Dg_(1,0,0) = 0

Dg_(1,1,1) = 144

Dg_(-1,-1,-1) = 240