Jacobian Matrix

analysis linear-algebra

Definition

Jacobian Matrix

Let be an open set, and let

be a totally differentiable function. The Jacobian matrix of at is the matrix

Its -entry is

The Jacobian matrix is the matrix representation of the derivative of at .

Linear approximation

Near , the function is approximated by its first-order linear part:

So the Jacobian describes how small changes in the input produce first-order changes in the output.

Dimensions

If , then:

• rows correspond to output coordinates
• columns correspond to input variables

Chain rule

If

then the chain rule can be written in matrix form as:

So composition corresponds to matrix multiplication of Jacobians.

Special cases

Scalar-valued function

If , then the Jacobian is a row matrix:

With the convention that the gradient is a column vector, this gives

One-dimensional input

If , then the Jacobian is an column matrix:

Examples

Function from to

Let

Then

and

Therefore,

The first row describes how the first output coordinate changes, and the second row describes how the second output coordinate changes.