Jacobian matrix
This article describes an analogue for functions of multiple variables of the following term/fact/notion for functions of one variable: derivative
Contents
Importance
The Jacobian matrix is the appropriate notion of derivative for a function that has multiple inputs (or equivalently, vector-valued inputs) and multiple outputs (or equivalently, vector-valued outputs).
Definition at a point
Direct epsilon-delta definition
Definition at a point in terms of gradient vectors as row vectors
Suppose is a vector-valued function with
-dimensional inputs and
-dimensional outputs. Explicitly, suppose
is a function with inputs
and outputs
. Suppose
is a point in the domain of
such that
is differentiable at
for
. Then, the Jacobian matrix of
at
is a
matrix of numbers whose
row is given by the gradient vector of
at
.
Explicitly, in terms of rows, it looks like:
Definition at a point in terms of partial derivatives
Suppose is a vector-valued function with
-dimensional inputs and
-dimensional outputs. Explicitly, suppose
is a function with inputs
and outputs
. Suppose
is a point in the domain of
such that
is differentiable at
for
. Then, the Jacobian matrix of
at
is a
matrix of numbers whose
entry is given by:
Here's how the matrix looks:
Note that for this definition to be correct, it is still necessary that the gradient vectors exist. If the gradient vectors do not exist but the partial derivatives do, a matrix can still be constructed using this recipe but it may not satisfy the nice behavior that the Jacobian matrix does.
Definition as a function
Definition in terms of gradient vectors as row vectors
Suppose is a vector-valued function with
-dimensional inputs and
-dimensional outputs. Explicitly, suppose
is a function with inputs
and outputs
. Then, the Jacobian matrix of
is a
matrix of functions whose
row is given by the gradient vector of
. Explicitly, it looks like this:
Note that the domain of this function is the set of points at which all the s individually are differentiable.
Definition in terms of partial derivatives
Suppose is a vector-valued function with
-dimensional inputs and
-dimensional outputs. Explicitly, suppose
is a function with inputs
and outputs
. Then, the Jacobian matrix of
is a
matrix of functions whose
entry is given by:
wherever all the s individually are differentiable in the sense of the gradient vectors existing. Here's how the matrix looks:
If the gradient vectors do not exist but the partial derivatives do, a matrix can still be constructed using this recipe but it may not satisfy the nice behavior that the Jacobian matrix does.
Particular cases
Case | What happens in that case? |
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The Jacobian matrix is the same as the matrix describing ![]() ![]() ![]() |
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The Jacobian matrix can then be thought of as a linear self-map from the ![]() |