Hessian Matrix Calculator

To calculate the hessian matrix, select the number of variables, enter the required values, and hit calculate button using hessian matrix calculator


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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.


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.


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).


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

f(x,y,z) = 2yx+ 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

fxx = 24x2y
fxy = 8x3
fxz = 6z2
fxy = 8 x^3
fyy = 12y
fyz = 0
fxz = 6z2
fyz = 0
fzz =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

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