Jacobian matrix

Summary

• \(f\) is a vector valued function of \(m\) variables \(x={\left( x_1, \ldots ,x_m \right)}'\) , \(f:R^m→R^n\) , \(y=f\left( x \right)\) where \(y={\left( y_1, \ldots ,y_n \right)}'\) .
• We write \(y_i=f_i\left( x \right)\) where \(f:R^m→R\) for \(i=1, \ldots ,n\)

\[y=\begin{bmatrix}y_1 \\ ⋮ \\ y_n\end{bmatrix}=\begin{bmatrix}f_1\left( x \right) \\ ⋮ \\ f_n\left( x \right)\end{bmatrix}=f\left( x \right)\]

• Example: \(f:R^2→R^2\) is defined by

\[y=f\left( x \right)=\begin{bmatrix}y_1 \\ y_2\end{bmatrix}=\begin{bmatrix}f_1\left( x \right) \\ f_2\left( x \right)\end{bmatrix}=\begin{bmatrix}2x_1+x_2^2 \\ 2x_2+x_1^2\end{bmatrix}\]

• Then, for example,

\[f\left( 1,0 \right)=y=\begin{bmatrix}y_1 \\ y_2\end{bmatrix}=\begin{bmatrix}2 \\ 1\end{bmatrix}\]

• We define the Jacobian \(J\) of \(f\) as the \(n×m\) matrix

\[J=\begin{bmatrix} \frac{∂f}{∂x_1} & \ldots & \frac{∂f}{∂x_m}\end{bmatrix}=\begin{bmatrix} \frac{∂f_1}{∂x_1} & \ldots & \frac{∂f_1}{∂x_m} \\ ⋮ & ⋱ & ⋮ \\ \frac{∂f_n}{∂x_1} & \ldots & \frac{∂f_n}{∂x_m}\end{bmatrix}\]

• that is

\[J_{i,j}= \frac{∂f_i}{∂x_j}\]

Example: \(f\) is defined as above. Then

\[J= \frac{∂f}{∂x}=\begin{bmatrix} \frac{∂f_1}{∂x_1} & \frac{∂f_1}{∂x_2} \\ \frac{∂f_2}{∂x_1} & \frac{∂f_2}{∂x_2}\end{bmatrix}=\begin{bmatrix}2 & 2x_2 \\ 2x_1 & 2\end{bmatrix}\]

Example: If \(A\) is an arbitrary \(n×m\) matrix and \(f\left( x \right)=Ax\) then

\[J= \frac{∂f}{∂x}=A\]