Jacobian¶

Formula¶

\[ J_f(x)=\left[\frac{\partial f_i}{\partial x_j}\right]_{i,j} \]

Parameters¶

\(f:\mathbb{R}^n\to\mathbb{R}^m\): vector-valued function
\(J_f(x)\in\mathbb{R}^{m\times n}\): Jacobian matrix

What it means¶

The Jacobian is the matrix of first-order partial derivatives for a vector-valued function.

What it's used for¶

Linearization of nonlinear maps.
Multivariable chain rule and backpropagation.

Key properties¶

Generalizes the gradient (for \(m=1\)).
Locally approximates \(f(x+\Delta x)\approx f(x)+J_f(x)\Delta x\).

Common gotchas¶

Different communities use transposed conventions.
Shape mismatches are common in implementation.

Example¶

If \(f(x,y)=(x+y,xy)\), then \(J_f=\begin{bmatrix}1&1\\ y&x\end{bmatrix}\).

How to Compute (Pseudocode)¶

Input: vector function f=(f1,...,fm): R^n -> R^m, point x
Output: Jacobian matrix J of size m x n

for i from 1 to m:
  for j from 1 to n:
    J[i,j] <- partial derivative of f_i with respect to x_j evaluated at x

return J

Complexity¶

Time: \(O(mn)\) partial-derivative evaluations at a high level
Space: \(O(mn)\) to store the Jacobian matrix
Assumptions: Excludes the internal cost of each derivative evaluation; structured/autodiff methods may compute Jacobian-vector products more efficiently than forming the full matrix