Jacobian matrix

This article deals with Jacobian matrices in analysis; for Jacobi matrices in operator theory, see Jacobi operator .

The Jacobian matrix (named after Carl Gustav Jacob Jacobi ; also functional matrix , derivative matrix , or Jacobian called) of a differentiable function is the - matrix of all the first partial derivatives . In the case of total differentiability , it forms the matrix representation of the first derivative of the function with respect to the standard bases des and des, which is understood as a linear mapping . ${\displaystyle f\colon {\mathbb {R} ^{n}}\to {\mathbb {R} ^{m}}\,\!}$ ${\displaystyle m\times n}$ ${\displaystyle f}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{m}}$

The Jacobi matrix is ​​used, for example, for approximate calculation (approximation) or minimization of multi-dimensional functions in mathematics .

definition

Let be a function whose component functions are denoted by and whose partial derivatives should all exist. For a point in space in the archetype space, let the respective associated coordinates. ${\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ ${\displaystyle f_{1},\ldots ,f_{m}}$ ${\displaystyle x}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle x_{1},\dots ,x_{n}}$

Then the point is through for the Jacobi matrix ${\displaystyle a\in U}$ ${\displaystyle a}$

${\displaystyle J_{f}(a):=\left({\frac {\partial f_{i}}{\partial x_{j}}}(a)\right)_{i=1,\ldots ,m;\ j=1,\ldots ,n}={\begin{pmatrix}{\frac {\partial f_{1}}{\partial x_{1}}}(a)&{\frac {\partial f_{1}}{\partial x_{2}}}(a)&\ldots &{\frac {\partial f_{1}}{\partial x_{n}}}(a)\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial f_{m}}{\partial x_{1}}}(a)&{\frac {\partial f_{m}}{\partial x_{2}}}(a)&\ldots &{\frac {\partial f_{m}}{\partial x_{n}}}(a)\end{pmatrix}}}$

Are defined.

The (transposed) gradients of the component functions of are in the lines of the Jacobi matrix . ${\displaystyle f_{1},\dots ,f_{m}}$ ${\displaystyle f}$

Other common realizations for the Jacobian matrix of at the site are , and . ${\displaystyle J_{f}(a)}$ ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle Df(a)}$ ${\displaystyle {\frac {\partial f}{\partial x}}(a)}$ ${\displaystyle \textstyle {\frac {\partial (f_{1},\ldots ,f_{m})}{\partial (x_{1},\ldots ,x_{n})}}(a)}$

example

The function is given by ${\displaystyle f\colon \mathbb {R} ^{3}\to \mathbb {R} ^{2}}$

${\displaystyle f(x,y,z)={\binom {x^{2}+y^{2}+z\cdot \sin x}{z^{2}+z\cdot \sin y}}}$

Then

${\displaystyle {\begin{aligned}{\frac {\partial }{\partial x}}f(x,y,z)&={\binom {2x+z\cdot \cos x}{0}}\\{\frac {\partial }{\partial y}}f(x,y,z)&={\binom {2y}{z\cdot \cos y}}\\{\frac {\partial }{\partial z}}f(x,y,z)&={\binom {\sin x}{2z+\sin y}}\end{aligned}}}$

and with it the Jacobi matrix

${\displaystyle J_{f}(x,y,z)=\left({\begin{array}{ccc}2x+z\cdot \cos x&2y&\sin x\\0&z\cdot \cos y\,&2z+\sin y\end{array}}\right)}$

Applications

• If the function is totally differentiable , then its total differential at that point is the linear mapping ${\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ ${\displaystyle Df_{a}}$ ${\displaystyle a=(a_{1},\dots ,a_{n})\in U}$

${\displaystyle Df_{a}\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m},\quad Df_{a}(h)=J_{f}(a)\cdot h}$.

The Jacobi matrix at this point is therefore the mapping matrix of . ${\displaystyle a}$ ${\displaystyle Df_{a}}$

• For the Jacobi matrix corresponds to the transposed gradient of . Sometimes the gradient is also defined as a line vector. In this case, the gradient and Jacobian matrix are the same. ${\displaystyle m=1}$ ${\displaystyle f}$

• The Jacobi matrix, if it is calculated for one place , can be used to approximate the function values of in the vicinity of : ${\displaystyle a=(a_{1},\dots ,a_{n})}$ ${\displaystyle f}$ ${\displaystyle a}$

${\displaystyle f(x)\approx f(a)+J_{f}(a)\cdot (x-a).}$

This affine mapping corresponds to the first order Taylor approximation ( linearization ).

• The propagation of measurement errors in the form of a covariance matrix occurs through the Jacobi matrix: ${\displaystyle V_{f}=J\cdot V_{x}\cdot J^{\text{T}}}$

Determinant of the Jacobian matrix

→ Main article : Jacobian determinant

Let us consider a differentiable function . Then its Jacobi matrix at the point is a square matrix. In this case one can determine the determinant of the Jacobi matrix . The determinant of the Jacobian matrix is ​​called the Jacobian determinant or functional determinant. If the Jacobi determinant is not equal to zero at the point , the function can be inverted in a neighborhood of . This is what the theorem of reverse mapping says . In addition, the Jacobian determinant plays an important role in the transformation theorem for integrals . If , by definition, one cannot form a determinant of the -Jacobi matrix. However, there is a similar concept in this case. This is called the Gram determinant . ${\displaystyle m=n}$ ${\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ ${\displaystyle J_{f}(a)}$ ${\displaystyle a\in U}$ ${\displaystyle n\times n}$ ${\displaystyle \det(J_{f}(a))}$ ${\displaystyle a}$ ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle m\neq n}$ ${\displaystyle m\times n}$

Jacobi matrix of a holomorphic function

In addition to functions , functions can also be examined for (complex) differentiability. Functions that are complexly differentiable are called holomorphic , because they have different properties than the (real) differentiable functions. Jacobi matrices can also be determined for the holomorphic function . There are two different variants here. On the one hand with complex-valued entries and on the other hand a matrix with real-valued entries. The -Jacobi matrix at the point is through ${\displaystyle f\colon U\subset \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ ${\displaystyle h\colon V\subset \mathbb {C} ^{n}\to \mathbb {C} ^{m}}$ ${\displaystyle h:=(h_{1},\ldots ,h_{m})\colon V\subset \mathbb {C} ^{n}\to \mathbb {C} ^{m}}$ ${\displaystyle m\times n}$ ${\displaystyle 2m\times 2n}$ ${\displaystyle m\times n}$ ${\displaystyle J_{h}^{\mathbb {C} }(z)}$ ${\displaystyle z:=(z_{1},\ldots ,z_{n})\in V\subset \mathbb {C} ^{n}}$

${\displaystyle J_{h}^{\mathbb {C} }(z):={\begin{pmatrix}{\frac {\partial h_{1}(z)}{\partial z_{1}}}&\cdots &{\frac {\partial h_{1}(z)}{\partial z_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial h_{m}(z)}{\partial z_{1}}}&\cdots &{\frac {\partial h_{m}(z)}{\partial z_{n}}}\end{pmatrix}}}$

Are defined.

Every complex-valued function can be split into two real-valued functions. That is, there are functions such that . The functions and can now usually be partially differentiated and arranged in a matrix. Be the coordinates in and set for everyone . The -Jacobi matrix of the holomorphic function at the point is then defined by ${\displaystyle u,v\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ ${\displaystyle h=u+iv}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle z:=(z_{1},\ldots ,z_{n})}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle z_{j}:=x_{j}+iy_{j}}$ ${\displaystyle j}$ ${\displaystyle 2m\times 2n}$ ${\displaystyle J_{h}^{\mathbb {R} }(z)}$ ${\displaystyle h}$ ${\displaystyle z\in V}$

${\displaystyle J_{h}^{\mathbb {R} }(z):={\begin{pmatrix}{\frac {\partial u_{1}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial u_{1}(z)}{\partial x_{n}}}&{\frac {\partial u_{1}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial u_{1}(z)}{\partial y_{n}}}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\{\frac {\partial u_{m}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial u_{m}(z)}{\partial x_{n}}}&{\frac {\partial u_{m}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial u_{m}(z)}{\partial y_{n}}}\\{\frac {\partial v_{1}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial v_{1}(z)}{\partial x_{n}}}&{\frac {\partial v_{1}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial v_{1}(z)}{\partial y_{n}}}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\{\frac {\partial v_{m}(z)}{\partial x_{1}}}&\cdots &{\frac {\partial v_{m}(z)}{\partial x_{n}}}&{\frac {\partial v_{m}(z)}{\partial y_{1}}}&\cdots &{\frac {\partial v_{m}(z)}{\partial y_{n}}}\end{pmatrix}}}$.

If the Jacobi matrices hold for holomorphic functions , one can of course consider the determinants of the two matrices. These two determinants are related to each other. It is true ${\displaystyle m=n}$

${\displaystyle \det \left(J_{h}^{\mathbb {R} }(z)\right)=\left|\det(J_{h}^{\mathbb {C} }(z))\right|^{2}}$.

See also

• Multi-dimensional chain rule
• Hessian matrix

literature

• Konrad Königsberger : Analysis 2. Springer-Verlag, Berlin / Heidelberg, 2000, ISBN 3-540-43580-8 . (for Jacobi matrices of real functions).
• Klaus Fritzsche, Hans Grauert : From Holomorphic Functions to Complex Manifolds . Springer-Verlag, ISBN 0-387-95395-7 . (Pp. 30–31; for Jacobi matrices of holomorphic functions).