Documentation

jacobian

Jacobian matrix

Syntax

Description

example

jacobian(f,v) computes the Jacobian matrix of f with respect to v. The (i,j) element of the result is f(i)v(j).

Examples

Jacobian of a Vector Function

The Jacobian of a vector function is a matrix of the partial derivatives of that function.

Compute the Jacobian matrix of [x*y*z, y^2, x + z] with respect to [x, y, z].

syms x y z
jacobian([x*y*z, y^2, x + z], [x, y, z])
ans =
[ y*z, x*z, x*y]
[   0, 2*y,   0]
[   1,   0,   1]

Now, compute the Jacobian of [x*y*z, y^2, x + z] with respect to [x; y; z].

jacobian([x*y*z, y^2, x + z], [x; y; z])

Jacobian of a Scalar Function

The Jacobian of a scalar function is the transpose of its gradient.

Compute the Jacobian of 2*x + 3*y + 4*z with respect to [x, y, z].

syms x y z
jacobian(2*x + 3*y + 4*z, [x, y, z])
ans =
[ 2, 3, 4]

Now, compute the gradient of the same expression.

gradient(2*x + 3*y + 4*z, [x, y, z])
ans =
 2
 3
 4

Jacobian with Respect to a Scalar

The Jacobian of a function with respect to a scalar is the first derivative of that function. For a vector function, the Jacobian with respect to a scalar is a vector of the first derivatives.

Compute the Jacobian of [x^2*y, x*sin(y)] with respect to x.

syms x y
jacobian([x^2*y, x*sin(y)], x)
ans =
  2*x*y
 sin(y)

Now, compute the derivatives.

diff([x^2*y, x*sin(y)], x)
ans =
[ 2*x*y, sin(y)]

Input Arguments

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f — Scalar or vector functionsymbolic expression | symbolic function | symbolic vector

Scalar or vector function, specified as a symbolic expression, function, or vector. If f is a scalar, then the Jacobian matrix of f is the transposed gradient of f.

v — Vector of variables with respect to which you compute Jacobiansymbolic variable | symbolic vector

Vector of variables with respect to which you compute Jacobian, specified as a symbolic variable or vector of symbolic variables. If v is a scalar, then the result is equal to the transpose of diff(f,v). If v is an empty symbolic object, such as sym([]), then jacobian returns an empty symbolic object.

More About

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Jacobian Matrix

The Jacobian matrix of the vector function f = (f1(x1,...,xn),...,fn(x1,...,xn)) is the matrix of the derivatives of f:

J(x1,xn)=[f1x1f1xnfnx1fnxn]

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