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 .
![f \ colon {\ mathbb {R} ^ {n}} \ to {\ mathbb {R} ^ {m}} \, \!](https://wikimedia.org/api/rest_v1/media/math/render/svg/7be19350296d356bd743520627d404ae62652611)
![m \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
![\ mathbb {R} ^ {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a87a024931038d1858dc22e8a194e5978c3412e)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2703b75767ff6fb23cc3d6ad1a25830d35eb5038)
![\ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
![x_ {1}, \ dots, x_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753)
Then the point is through
for the Jacobi matrix![a \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1991ea9cb2ab076462a5538242321f0d0ee991)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6913db61c168b3ab510854dd2bb6d33ec564ccc0)
Are defined.
The (transposed) gradients of the component functions of are in the lines of the Jacobi matrix .
![f_ {1}, \ dots, f_ {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f067fd333142a9525e2bb6d75e68879724992b)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
Other common realizations for the Jacobian matrix of at the site are , and .
![J_ {f} (a)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d602622c85d01faa032fe267d343360ccbdf98fd)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![Df (a)](https://wikimedia.org/api/rest_v1/media/math/render/svg/17c83c98b9b68b09db509888726883bae6a68aba)
![{\ displaystyle {\ frac {\ partial f} {\ partial x}} (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/744dbba1af9277b34cc5aa48c637eae4389b33a6)
![{\ displaystyle \ textstyle {\ frac {\ partial (f_ {1}, \ ldots, f_ {m})} {\ partial (x_ {1}, \ ldots, x_ {n})}} (a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85d4f41511642601b0d8b389b20f46d1935712ac)
example
The function
is given by
![{\ displaystyle f \ colon \ mathbb {R} ^ {3} \ to \ mathbb {R} ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13bb78d3317a6a897796ca9023bcd7e22ab00fb5)
![f (x, y, z) = {\ binom {x ^ {2} + y ^ {2} + z \ cdot \ sin x} {z ^ {2} + z \ cdot \ sin y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/201eef5d272f72d8ae8db77d2c4c439469740412)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9be09b4c2d09d7aba35d4929c70d7175a49cd96)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/806f1a19d6f7bfd667bea48a52f0938ffe654e36)
Applications
- If the function is totally differentiable , then its total differential at that point is the linear mapping
![{\ displaystyle Df_ {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32a8678a7f42b6144a93bad7cd7890d4b732c410)
![a = (a_ {1}, \ dots, a_ {n}) \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/f061d58e1e14decdd2958a2b0b7e90b61b995fe0)
-
.
- The Jacobi matrix at this point is therefore the mapping matrix of .
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![{\ displaystyle Df_ {a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32a8678a7f42b6144a93bad7cd7890d4b732c410)
- 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.
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
- 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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0507c57a117ff1a5c9eb8823880fefeab15270f)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![{\ displaystyle f (x) \ approx f (a) + J_ {f} (a) \ cdot (xa).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/947e0d845f39e4c6dd988231b7d1130cb71cdf2d)
- This affine mapping corresponds to the first order Taylor approximation ( linearization ).
Determinant of the Jacobian matrix
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 .
![m = n](https://wikimedia.org/api/rest_v1/media/math/render/svg/69c9d8e54796e7de7d4738510cc10bc3fc55d48e)
![f \ colon U \ subset \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2b61ed55f4015515ede337fde29c44f68cccb5)
![J_ {f} (a)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d602622c85d01faa032fe267d343360ccbdf98fd)
![a \ in U](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1991ea9cb2ab076462a5538242321f0d0ee991)
![n \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78)
![\ det (J_ {f} (a))](https://wikimedia.org/api/rest_v1/media/math/render/svg/8116c75b42d68db7570ecfcb004c86729bb163f8)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![m \ neq n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a50a2eaa8dcad2a8e19c9fd861a0fdd641bdfa46)
![m \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2703b75767ff6fb23cc3d6ad1a25830d35eb5038)
![{\ displaystyle h \ colon V \ subset \ mathbb {C} ^ {n} \ to \ mathbb {C} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c54aea5a96cf04363934e5d543360560ccff1cee)
![{\ displaystyle h: = (h_ {1}, \ ldots, h_ {m}) \ colon V \ subset \ mathbb {C} ^ {n} \ to \ mathbb {C} ^ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9a89daeaf810495c4bd22ba07bdf3aa8d6f87b2)
![m \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d)
![2m \ times 2n](https://wikimedia.org/api/rest_v1/media/math/render/svg/6467a4f277236af1ac041678961d766d610678a2)
![m \ times n](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d)
![{\ displaystyle J_ {h} ^ {\ mathbb {C}} (z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d93598c76e8905dab571f383e87749dc047218)
![{\ displaystyle z: = (z_ {1}, \ ldots, z_ {n}) \ in V \ subset \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0099d97031a9899606dc3a4cb0c6c4feb204d0c8)
![{\ 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} (e.g. )} {\ partial z_ {1}}} & \ cdots & {\ frac {\ partial h_ {m} (z)} {\ partial z_ {n}}} \ end {pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d48c63f058f2b5b9d6d9b43f37c8a380b4017cd6)
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
![u, v \ colon \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9810f1211ab0e1f4723c329d0942275b8e32fc80)
![h = u + iv](https://wikimedia.org/api/rest_v1/media/math/render/svg/918d925a4e29199f97f6246cdf538a1f0e3ba2c9)
![u](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![z: = (z_ {1}, \ ldots, z_ {n})](https://wikimedia.org/api/rest_v1/media/math/render/svg/95ee2bf6b9ac9831e1a66df33078d31b55f50086)
![{\ displaystyle \ mathbb {C} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258)
![z_ {j}: = x_ {j} + iy_ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8221b88aadd092db5d6f3d45d21b564405866bf)
![j](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0)
![2m \ times 2n](https://wikimedia.org/api/rest_v1/media/math/render/svg/6467a4f277236af1ac041678961d766d610678a2)
![J_ {h} ^ {\ mathbb {R}} (z)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9727c2ea4e4f96ff11df1a669dbaed1f7c5854d)
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
![z \ in V](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e530fb380ba488705c57df90a2631e9055b7e7)
-
.
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
![m = n](https://wikimedia.org/api/rest_v1/media/math/render/svg/69c9d8e54796e7de7d4738510cc10bc3fc55d48e)
-
.
See also
literature