## Introduction to Jacobian Matlab

Jacobian is a determinant or defined for a finite number of input functions and the same finite number of variables. Each row of Jacobian will consist of 1^{st} partial derivatives of the input function w.r.t each variable. In this topic, we are going to learn about Jacobian Matlab.

In MATLAB, Jacobian is mainly of 2 types:

- Vector function’s Jacobian: It is a matrix with partial derivatives of the input vector function
- Scalar function’s Jacobian: For a scalar function, Jacobian gives transpose of the input function’s gradient

We use the Jacobian function in MATLAB to get the Jacobian matrix.

**Syntax:**

`jacobian (F, Z)`

**Description:**

jacobian (F, Z) is used to get the Jacobian matrix for input function ‘F’ w.r.t Z.

### Examples of Jacobian Matlab

Let us now understand the code to get the Jacobian matrix in MATLAB using different examples:

#### Example #1

In this example, we will take a vector function and will compute its Jacobian Matrix using the Jacobian function. For our first example, we will input the following values:

- Pass the input vector function as [b*a, a + c, b^3]
- Pass the variables as [a, b, c]

**Code:**

`syms a b c`

[Initializing the variables ‘a’, ‘b’, ‘c’]
jacobian ([b*a, a+c, b^3], [a, b, c])

[Passing the input vector function and variables as the arguments]
* Mathematically, the Jacobian matrix of [b*a, a+c, b^3] concerning [a, b, c] is

[ b, a, 0]
[ 1, 0, 1]
[ 0, 3*b^2, 0]

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**Input:**

`syms a b c`

jacobian ([b*a, a+c, b^3], [a, b, c])

**Output**

As we can see in the output, we have obtained partial derivative of every element of input vector function w.r.t each variable passed as input.

#### Example #2

In this example, we will take another vector function and will compute its Jacobian Matrix using the Jacobian function. For this example, we will input following values:

- Pass the input vector function as [a^4 + b, a^2 + c, b + 3]
- Pass the variables as [a, b, c]

**Code:**

`syms a b c`

[Initializing the variables ‘a’, ‘b’, ‘c’]
jacobian([a^4 + b, a^2 + c, b + 3], [a, b, c])

[Passing the input vector function and variables as the arguments]
* Mathematically, the Jacobian matrix of [a^4 + b, a^2 + c, b + 3]with respect to [a, b, c] is

[ 4*a^3, 1, 0]
[ 2*a, 0, 1]
[ 0, 1, 0]

**Input:**

`syms a b c`

jacobian([a^4 + b, a^2 + c, b + 3], [a, b, c])

**Output:**

As we can see in the output, we have obtained partial derivative of every element of input vector function w.r.t each variable passed as input.

#### Example #3

In this example, we will take another vector function and will compute its Jacobian Matrix using the Jacobian function. For this example, we will input following values:

- Pass the input vector function as [c^3 + b^2, 2*a^3 + c, c^2 + 5]
- Pass the variables as [a, b, c]

**Code:**

`syms a b c`

[Initializing the variables ‘a’, ‘b’, ‘c’]
jacobian([c^3 + b^2, 2*a^3 + c, c^2 + 5], [a, b, c])

[Passing the input vector function and variables as the arguments]
* Mathematically, the Jacobian matrix of [c^3 + b^2, 2*a^3 + c, c^2 + 5] with respect to [a, b, c] is

[ 0, 2*b, 3*c^2]
[ 6*a^2, 0, 1]
[ 0, 0, 2*c]

**Input:**

`syms a b c`

jacobian([c^3 + b^2, 2*a^3 + c, c^2 + 5], [a, b, c])

**Output:**

As we can see in the output, we have obtained partial derivative of every element of input vector function w.r.t each variable passed as input.

Next, let us take a few examples to understand the Jacobian function in the case of Scalars.

#### Example #4

In this example, we will take a scalar function and will compute its Jacobian Matrix using the Jacobian function. For our first example, we will input the following values:

- Pass the input scalar function as 3*a^2 + c*b^3 + a^2
- Pass the variables as [a, b, c],

**Code:**

`syms a b c`

[Initializing the variables ‘a’, ‘b’, ‘c’]
jacobian(3*a^2 + c*b^3 + a^2, [a, b, c])

[Passing the input scalar function and variables as the arguments]
* Mathematically, the Jacobian matrix of 3*a^2 + c*b^3 + a^2 concerning [a, b, c] is

[ 8*a, 3*b^2*c, b^3, 0]

**Input:**

`syms a b c`

jacobian(3*a^2 + c*b^3 + a^2, [a, b, c])

**Output:**

As we can see in the output, we have obtained transpose of the gradient as the Jacobian matrix for a scalar function.

#### Example #5

In this example, we will take another scalar function and will compute its Jacobian Matrix using the Jacobian function. For our first example, we will input the following values:

- Pass the input scalar function as b^4 + a^3 – c*d
- Pass the variables as [a, b, c, d],

**Code:**

`syms a b c d`

[Initializing the variables ‘a’, ‘b’, ‘c’, ‘d’]
jacobian(b^4 + a^3 - c*d, [a, b, c, d])

[Passing the input scalar function and variables as the arguments]
* Mathematically, the Jacobian matrix of b^4 + a^3 -c*d concerning [a, b, c, d] is

[ 3*a^2, 4*b^3, -d, -c]

**Input:**

`syms a b c d`

jacobian(b^4 + a^3 - c*d, [a, b, c, d])

**Output:**

As we can see in the output, we have obtained transpose of the gradient as the Jacobian matrix for a scalar function.

### Conclusion

The jacobian function is used in MATLAB to find the Jacobian matrix of any function (vector or scalar). For a scalar, the Jacobian function provides us with the transpose of the gradient for the scalar function.

### Recommended Articles

This is a guide to Jacobian Matlab. Here we discuss the Jacobian matrix in MATLAB using different examples along with the sample codes. You may also have a look at the following articles to learn more –