Jacobian matrix and determinant.pdf

Jacobian matrix and determinant 1 Jacobian matrix

In vector calculus, the Jacobian matrix (/dʒɨˈkoʊbiən/, /jɨˈkoʊbiən/) is the matrix of all first-order partial derivatives of a vector-valued function. Specifically, suppose f : ℝn → ℝm is a function which takes as input the vector x ∈ ℝn and produces as output the vector f(x) ∈ ℝm . Then the Jacobian matrix J of f is an m×n matrix, usually defined and arranged as follows: 

J=

[ df ∂f = dx ∂x1

···

∂f1  ]  ∂x. 1 ∂f =  .. ∂xn  ∂f m ∂x1

··· ..

.

···

 ∂f1 ∂xn  ..   .  ∂fm  ∂xn

If a function is differentiable at a point, its derivative is given in coordinates by the Jacobian, but a function doesn't need to be differentiable for the Jacobian to be defined, since only the partial derivatives are required to exist.

or, component-wise:

Ji,j =

The Jacobian generalizes the gradient of a scalar-valued function of multiple variables, which itself generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian for a scalar-valued multivariable function is the gradient and that of a scalar-valued function of single variable is simply its derivative. The Jacobian can also be thought of as describing the amount of “stretching”, “rotating” or “transforming” that a transformation imposes locally. For example, if (x′, y′) = f(x, y) is used to transform an image, the Jacobian J (x, y), describes how the image in the neighborhood of (x, y) is transformed.

∂fi . ∂xj

If p is a point in ℝn and f is differentiable at p, then its derivative is given by J (p). In this case, the linear map This matrix, whose entries are functions of x, is also de- described by J (p) is the best linear approximation of f noted by Df, J , and ∂(f1 ,...,f )/∂(x1 ,...,x ). (Note that near the point p, in the sense that some literature defines the Jacobian as the transpose of the matrix given above.) The Jacobian matrix is important because if the function f(x) = f(p) + Jf (p)(x − p) + o(∥x − p∥) f is differentiable at a point x (this is a slightly stronger for x close to p and where o is the little o-notation (for x condition than merely requiring that all partial derivatives → p) and ‖x − p‖ is the distance between x and p. exist there), then the Jacobian matrix defines a linear map ℝn → ℝm , which is the best linear approximation of the Compare this to a Taylor series for a scalar function of a function f near the point x. This linear map is thus the scalar argument, truncated to first order: generalization of the usual notion of derivative, and is called the derivative or the differential of f at x. f (x) = f (p) + f ′ (p)(x − p) + o(x − p). If m = n, the Jacobian matrix is a square matrix, and its determinant, a function of x1 , …, xn, is the Jacobian de- In a sense, both the gradient and Jacobian are "first derivaterminant of f. It carries important information about tives"—the former the first derivative of a scalar function the local behavior of f. In particular, the function f has of several variables, the latter the first derivative of a veclocally in the neighborhood of a point x an inverse func- tor function of several variables. tion that is differentiable if and only if the Jacobian de- The Jacobian of the gradient of a scalar function of sevterminant is nonzero at x (see Jacobian conjecture). The eral variables has a special name: the Hessian matrix, Jacobian determinant occurs also when changing the vari- which in a sense is the "second derivative" of the funcables in multi-variable integrals (see substitution rule for tion in question. multiple variables). If m = 1, f is a scalar field and the Jacobian matrix is reduced to a row vector of partial derivatives of f—i.e. the gradient of f.

2 Jacobian determinant

These concepts are named after the mathematician Carl If m=n, then f is a function from ℝn to itself and the JaGustav Jacob Jacobi (1804–1851). cobian matrix is a square matrix. We can then form its 1

2

5 EXAMPLES

f

A nonlinear map f : R2 → R2 sends a small square to a distorted parallelepiped close to the image of the square under the best linear approximation of f near the point.

The (unproved) Jacobian conjecture is related to global invertibility in the case of a polynomial functions, that is a function defined by n polynomials in n variables. It asserts that, if the Jacobian determinant is a non-zero constant (or, equivalently, that it does not have any complex zero), then the function is invertible and its inverse is a polynomial function.

4 Critical points Main article: Critical point

determinant, known as the Jacobian determinant. The Jacobian determinant is occasionally referred to as “the If f : ℝn → ℝm is a differentiable function, a critical point of f is a point where the rank of the Jacobian matrix is not Jacobian”. maximal. This means that the rank at the critical point is The Jacobian determinant at a given point gives imporlower than the rank at some neighbour point. In other tant information about the behavior of f near that point. words, let k be the maximal dimension of the open balls For instance, the continuously differentiable function f is contained in the image of f; then a point is critical if all n invertible near a point p ∈ ℝ if the Jacobian determinant minors of rank k of f are zero. at p is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at p is positive, In the case where 1 = m = n = k, a point is critical if the then f preserves orientation near p; if it is negative, F Jacobian determinant is zero. reverses orientation. The absolute value of the Jacobian determinant at p gives us the factor by which the function f expands or shrinks volumes near p; this is why it occurs 5 Examples in the general substitution rule. The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain. To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral. This is because the n-dimensional dV element is in general a parallelepiped in the new coordinate system, and the n-volume of a parallelepiped is the determinant of its edge vectors.

5.1 Example 1 Consider the function f : ℝ2 → ℝ2 given by [

] x2 y f(x, y) = . 5x + sin y Then we have

The Jacobian can also be used to solve systems of differential equations at an equilibrium point or approximate f1 (x, y) = x2 y solutions near an equilibrium point. and

3

Inverse

According to the inverse function theorem, the matrix in- f2 (x, y) = 5x + sin y verse of the Jacobian matrix of an invertible function is and the Jacobian matrix of F is the Jacobian matrix of the inverse function. That is, if the Jacobian of the function f : ℝn → ℝn is continuous and   nonsingular at the point p in ℝn , then f is invertible when ∂f1 ∂f1 [ restricted to some neighborhood of p and  ∂x ∂y   = 2xy Jf (x, y) =   ∂f2 ∂f2  5 −1 ∂x ∂y J −1 ◦ f = Jf . f

Conversely, if the Jacobian determinant is not zero at a and the Jacobian determinant is point, then the function is locally invertible near this point, that is there is neighbourhood of this point, in which the function is invertible. det(Jf (x, y)) = 2xy cos y − 5x2 .

x2 cos y

]

5.5

5.2

Example 5

3

Example 2: polar-Cartesian transformation

The transformation from polar coordinates (r, φ) to Cartesian coordinates (x, y), is given by the function F: ℝ+ × [0, 2 π) → ℝ2 with components:

y3 = 4x22 − 2x3 y4 = x3 sin x1 is

x = r cos φ; y = r sin φ. 

y1 = x1 y2 = 5x3



 ∂x [ ] ∂φ   = cos φ −r sin φ sin φ r cos φ ∂y  ∂φ

∂y1  ∂x1   ∂y2   ∂x  1 JF (x1 , x2 , x3 ) =   ∂y3  The Jacobian determinant is equal to r. This can be used  ∂x1  to transform integrals between the two coordinate sys ∂y4 tems: ∂x1 ∂x  ∂r J(r, φ) =   ∂y ∂r

∫∫

∫∫ f (r cos ϕ, r sin ϕ) r dr dϕ.

f (x, y) dx dy = A

5.3

∂y1 ∂x2 ∂y2 ∂x2 ∂y3 ∂x2 ∂y4 ∂x2

 ∂y1 ∂x3    ∂y2  1  ∂x3  0   = 0 ∂y3    x cos x1  3 ∂x3   ∂y4 ∂x3

0 0 8x2 0

 0 5  . −2  sin x1

This example shows that the Jacobian need not be a square matrix.

A

Example 3: spherical-Cartesian trans- 5.5 Example 5 formation

The Jacobian determinant of the function F : ℝ3 → ℝ3 The transformation from spherical coordinates (r, θ, φ) with components to Cartesian coordinates (x, y, z), is given by the function F: ℝ+ × [0, π] × [0, 2 π) → ℝ3 with components: y1 = 5x2 x = r sin θ cos φ; y = r sin θ sin φ; z = r cos θ. The Jacobian matrix for this coordinate change is

y2 = 4x21 − 2 sin(x2 x3 ) y3 = x2 x3 is 0 8x1 0

5 0 5 −2x3 cos(x2 x3 ) −2x2 cos(x2 x3 ) = −8x1   x3 ∂x ∂x ∂x x3 x2  ∂r ∂θ ∂φ      sin θ cos φ r cos θ cos φ −r sin θ sin φ  ∂y ∂y ∂y     From this we see that F reverses orientation near those JF (r, θ, φ) =   = sin θ sin φ r cos θ sin φ r sin θ cos φ .  ∂r ∂θ ∂φ  points where x1 and x2 have the same sign; the function cos θ −r sin θ 0   is locally invertible everywhere except near points where  ∂z ∂z ∂z  x1 = 0 or x2 = 0. Intuitively, if one starts with a tiny object ∂r ∂θ ∂φ around the point (1, 2, 3) and apply F to that object, one The determinant is r2 sin θ. As an example, since dV = will get a resulting object with approximately 40 × 1 × 2 dx1 dx2 dx3 this determinant implies that the differential = 80 times the volume of the original one. volume element dV = r2 sin θ dr dθ dφ. Nevertheless this determinant varies with coordinates.

6 Other uses 5.4

Example 4

The Jacobian serves as a linearized design matrix in staThe Jacobian matrix of the function F : ℝ3 → ℝ4 with tistical regression and curve fitting; see non-linear least components squares.

0 = −40x1 x x2

4

10

6.1

Dynamical systems

Consider a dynamical system of the form x′ = F(x), where x′ is the (component-wise) time derivative of x, and F : ℝn → ℝn is differentiable. If F(x0 ) = 0, then x0 is a stationary point (also called a critical point; this is not to be confused with fixed points). The behavior of the system near a stationary point is related to the eigenvalues of JF(x0 ), the Jacobian of F at the stationary point.[1] Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point, if any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability.

6.2

Newton’s method

A system of coupled nonlinear equations can be solved iteratively by Newton’s method. This method uses the Jacobian matrix of the system of equations.

7

See also • Hessian matrix • Pushforward (differential)

8

References

[1] Arrowsmith, D. K.; Place, C. M. (1992). “Section 3.3”. Dynamical Systems. London: Chapman & Hall. ISBN 0-412-39080-9.

9

Further reading • Gandolfo, Giancarlo (1996). Economic Dynamics (Third ed.). Berlin: Springer. pp. 305–330. ISBN 3-540-60988-1.

10

External links

• Hazewinkel, Michiel, ed. (2001), “Jacobian”, Encyclopedia of Mathematics, Springer, ISBN 9781-55608-010-4 • Mathworld A more technical explanation of Jacobians

EXTERNAL LINKS

5

11 11.1

Text and image sources, contributors, and licenses Text

• Jacobian matrix and determinant Source: http://en.wikipedia.org/wiki/Jacobian%20matrix%20and%20determinant?oldid=659831051 Contributors: AxelBoldt, Jqt, Michael Hardy, Delirium, Looxix~enwiki, Kevin Baas, AugPi, Charles Matthews, Dysprosia, Jitse Niesen, Zero0000, Alembert~enwiki, Robbot, Josh Cherry, YahoKa, Fredrik, Rasmus Faber, Robinh, Isopropyl, Giftlite, BenFrantzDale, Everyking, Edrex, Vadmium, Leonard Vertighel, Karol Langner, Discospinster, ObsessiveMathsFreak, Bender235, Kwamikagami, Wood Thrush, Tsirel, Varuna, Jumbuck, Keenan Pepper, Voltagedrop, Oleg Alexandrov, Thryduulf, LizardWizard, Kzollman, Mpatel, Tabletop, Bluemoose, Rjwilmsi, HappyCamper, R.e.b., Bushido Hacks, FlaBot, Mathbot, Sanpaz, WriterHound, YurikBot, Wavelength, Dotancohen, Archelon, Pmdboi, Syth, Tong~enwiki, Snaxe920, Teply, Zvika, SmackBot, Eskimbot, BiT, PeterSymonds, Njerseyguy, Silly rabbit, Complexica, Nbarth, DHN-bot~enwiki, John Reaves, Bsilverthorn, Spakoj~enwiki, Rpf, Lambiam, Maverick starstrider, Jim.belk, Asyndeton, JoeBot, JKrabbe, Paul Matthews, Cydebot, Justincop, Mathmoclaire, SGGH, Batmanshotokan, Ben pcc, Joe Schmedley, JAnDbot, Deflective, MSBOT, VoABot II, Stevvers, User A1, J.delanoy, Melink14, LordAnubisBOT, Policron, Fylwind, LokiClock, Rei-bot, Cloudswrest, Dch888, Geometry guy, Graymornings, Jhawkinson, Scottkosty, Ivan Štambuk, YonaBot, Cwkmail, Yintan, Smarchesini, OKBot, Blacklemon67, Randomblue, Denisarona, UKoch, Rubybrian, Qwfp, Tomhosking, MystBot, CàlculIntegral, Addbot, Fgnievinski, Topology Expert, EconoPhysicist, TStein, Ozob, Wikomidia, Leycec, Luckas-bot, Garymm, , Hebr138, NickK, Kike123unc, ArthurBot, Bdmy, Etoombs, DBAllan, GrouchoBot, BradLipovsky, Point-set topologist, Kristjan.Jonasson, TobeBot, Wcedric, Jangekheid, RjwilmsiBot, 123Mike456Winston789, Sweethaws, EmausBot, Gpooseh, D.Lazard, Mirko.raca, U+003F, IznoRepeat, Heinzhill, ClueBot NG, PGulf, Neojamaj, Dayson39, Anelson0528, ChrisGualtieri, Makecat-bot, Mihaif7, CsDix, Zoydb, Mi scot87, AioftheStorm, Ingmar Schuster, ArchimedesZ, Phleg1, JBrobst, Hayazin, Gholleywiki, Brownzb and Anonymous: 158

11.2

Images

• File:Jacobian_determinant_and_distortion.svg Source: http://upload.wikimedia.org/wikipedia/en/9/96/Jacobian_determinant_and_ distortion.svg License: CC-BY-SA-3.0 Contributors: created in inkscape Original artist: Blacklemon67

11.3

Content license

• Creative Commons Attribution-Share Alike 3.0