Jacobian

來源
Given a set y=f(x) of n variables x1,...,xn , written explicitly as

y=⎡⎣⎢⎢⎢⎢⎢f1(x)f2(x)⋮fn(x)⎤⎦⎥⎥⎥⎥⎥

or more explicitly as

⎧⎩⎨⎪⎪y1=f1(x1,...,xn)⋮yn=fn(x1,...,xn)

the Jacobian matrix, sometimes simply called “the Jacobian”(Simon and Blume 1994) is defined by

J(x1,...,xn)=⎡⎣⎢⎢⎢⎢⎢⎢⎢∂y1∂x1⋮∂yn∂x1⋯⋱⋯∂y1∂xn⋮∂yn∂xn⎤⎦⎥⎥⎥⎥⎥⎥⎥

The determinant of J is the Jacobian determinant (confusingly, often called “the Jacobian” as well) and is denoted

J=∣∣∣∂(y1,...,yn)∂(x1,...,xn)∣∣∣

---

Practical 4

The Jacobian is a m×n matrix of derivatives for a multivariate function f:Rn→Rm :

dfdx=⎡⎣⎢⎢⎢⎢⎢⎢⎢∂f1∂x1⋮∂fm∂x1⋯⋱⋯∂f1∂xn⋮∂fm∂xn⎤⎦⎥⎥⎥⎥⎥⎥⎥

each i th row being a gradient of an element, fi , of the output vector, f .