Jacobian matrix and determinant

The Jacobian matrix is used for example for the approximate calculation ( approximation) or minimizing multidimensional functions in mathematics.

Definition

Let be a function whose partial derivatives all exist, with the component functions. Also, are denoted by the coordinates in the original image space. Of the Jacobian matrix is ​​at the point then

Or in detail by

Defined.

Example

The function is given by

Then

And thus the Jacobian

Applications

  • Is the total function differentiable, the Jacobian matrix is ​​a cross plot of the derivative.
  • Of the Jacobian matrix corresponding to the transpose of the gradient. Sometimes, the gradient is defined as a row vector. In this case, gradient and Jacobian matrix are equal.
  • The Jacobi matrix, when they calculates a point to approximate the function values ​​are used in the vicinity of: This affine transformation corresponds to the Taylor first order approximation (linearization ).
  • The propagation of measurement errors in the form of a covariance matrix of the Jacobian matrix is done by:

Determinant of the Jacobian matrix

Be, it is therefore considered to be a differentiable function. Then the Jacobian matrix at point a square matrix. In this case one can determine the determinant of the Jacobian matrix. The determinant of the Jacobian matrix is called the Jacobian or functional determinant. If the Jacobian at the point equal to zero, the function is invertible in a neighborhood of. This implies the theorem of the inverse map. In addition, the Jacobian plays an important role in the transformation rate of integrals. If so of course you can not form a determinant of the Jacobian matrix. However, there is a similar concept in this case. This is called the Gram determinant.

Jacobian of a holomorphic function

Auxiliary functions can be studied also functions on (complex) differentiable. Functions that are complex-differentiable are called holomorphic, because they have different properties than the ( real) differentiable functions. Also for the holomorphic function, one can determine Jacobian matrices. There are two different versions. Firstly, a complex-valued entries, and on the other a matrix with real-valued entries. The Jacobian matrix at the point of

Defined.

Each complex-valued function can be split into two real-valued functions. That is, there are features such that applies. The functions and one can again differentiate usually partial and arrange them in a matrix. Be the coordinates in and sit for all. The Jacobi matrix of holomorphic function at the point is then defined by

Applies to the Jacobian matrices for holomorphic functions, so you can of course look at the determinants of the two matrices. These two determinants are interrelated. It is namely

Total derivative Hessian matrix Hans Grauert
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