Jacobijeva matrika

Jacobijeva matrika (oznaka $J\,$$J\,$ ali $J_{f}(x_{1},\dots ,x_{n})\,$$J_{f}(x_{1},\dots ,x_{n})\,$) je matrika, ki jo sestavljajo parcialni odvodi prvega reda vektorja.

Determinanta, ki jo dobimo iz Jakobijeve matrike, se imenuje Jacobijeva determinanta.

Imenujeta se po nemškem matematiku Carlu Gustavu Jacobu Jacobiju (1804 – 1851).

Matrika ima obliko:

$J={\begin{bmatrix}{\dfrac {\partial y_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial y_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial y_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial y_{m}}{\partial x_{n}}}\end{bmatrix}}\,$$J={\begin{bmatrix}{\dfrac {\partial y_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial y_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial y_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial y_{m}}{\partial x_{n}}}\end{bmatrix}}\,$.

V matriki i-ta vrstica odgovarja gradientu i-te komponente funkcije $y_{i}\,$$y_{i}\,$ ali $\left(\nabla y_{i}\right)\,$$\left(\nabla y_{i}\right)\,$.

Determinanto kvadratne Jacobijeve matrike včasih imenujejo tudi jakobian . V literaturi se pogosto uporablja isti izraz tudi za transponirano matriko zgornje matrike.

Jacobijeva matrika

Če je dana preslikava $\mathbf {f} :\mathbb {R} ^{n}\to \mathbb {R} ^{m},\mathbf {f} =(f_{1},\ldots ,f_{m}),_{i}=f_{i}(x_{1},\ldots ,x_{n}),i=1,\ldots ,m\,$$\mathbf {f} :\mathbb {R} ^{n}\to \mathbb {R} ^{m},\mathbf {f} =(f_{1},\ldots ,f_{m}),_{i}=f_{i}(x_{1},\ldots ,x_{n}),i=1,\ldots ,m\,$ in so v neki točki $x\,$$x\,$ dani vsi prvi parcialni odvodi, potem je dana tudi Jacobijeva matrika razsežnosti $m\times n\,$$m\times n\,$. Jacobijeva matrika neke funkcije določa orientacijo tangentne ravnine na funkcijo v dani točki. Tako Jacobijeva matrika posplošuje gradient skalarne funkcije večjega števila spremenljivk.

Jacobijevo matriko označujemo z

$J_{f}\quad \quad \,$$J_{f}\quad \quad \,$ ali

$Df\quad \quad \,$$Df\quad \quad \,$ ali

$\textstyle {\frac {\partial f}{\partial x}}\quad \quad \,$$\textstyle {\frac {\partial f}{\partial x}}\quad \quad \,$ ali

$\textstyle {\frac {\partial (f_{1},\ldots ,f_{m})}{\partial (x_{1},\ldots ,x_{n})}}\quad \quad \,$$\textstyle {\frac {\partial (f_{1},\ldots ,f_{m})}{\partial (x_{1},\ldots ,x_{n})}}\quad \quad \,$.

Primer

Primer 1

Za primer poglejmo pretvorbo sfernih koordinat $(r,\theta ,\phi )\,$$(r,\theta ,\phi )\,$ v kartezični koordinatni sistem $(x_{1},x_{2},x_{3})\,$$(x_{1},x_{2},x_{3})\,$ pretvorba je dana s funkcijo $f:R^{+}\times [0,\pi ]\times [0,2\pi ]\to R^{3}\,$$f:R^{+}\times [0,\pi ]\times [0,2\pi ]\to R^{3}\,$ s komponentami

$x_{1}=r\,\sin \theta \,\cos \phi \,$$x_{1}=r\,\sin \theta \,\cos \phi \,$

$x_{2}=r\,\sin \theta \,\sin \phi \,$$x_{2}=r\,\sin \theta \,\sin \phi \,$

$x_{3}=r\,\cos \theta .\,$$x_{3}=r\,\cos \theta .\,$.

Jacobijeva matrika je

$J_{f}(r,\theta ,\phi )={\begin{bmatrix}{\dfrac {\partial x_{1}}{\partial r}}&{\dfrac {\partial x_{1}}{\partial \theta }}&{\dfrac {\partial x_{1}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{2}}{\partial r}}&{\dfrac {\partial x_{2}}{\partial \theta }}&{\dfrac {\partial x_{2}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{3}}{\partial r}}&{\dfrac {\partial x_{3}}{\partial \theta }}&{\dfrac {\partial x_{3}}{\partial \phi }}\\\end{bmatrix}}={\begin{bmatrix}\sin \theta \,\cos \phi &r\,\cos \theta \,\cos \phi &-r\,\sin \theta \,\sin \phi \\\sin \theta \,\sin \phi &r\,\cos \theta \,\sin \phi &r\,\sin \theta \,\cos \phi \\\cos \theta &-r\,\sin \theta &0\end{bmatrix}}.$$J_{f}(r,\theta ,\phi )={\begin{bmatrix}{\dfrac {\partial x_{1}}{\partial r}}&{\dfrac {\partial x_{1}}{\partial \theta }}&{\dfrac {\partial x_{1}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{2}}{\partial r}}&{\dfrac {\partial x_{2}}{\partial \theta }}&{\dfrac {\partial x_{2}}{\partial \phi }}\\[3pt]{\dfrac {\partial x_{3}}{\partial r}}&{\dfrac {\partial x_{3}}{\partial \theta }}&{\dfrac {\partial x_{3}}{\partial \phi }}\\\end{bmatrix}}={\begin{bmatrix}\sin \theta \,\cos \phi &r\,\cos \theta \,\cos \phi &-r\,\sin \theta \,\sin \phi \\\sin \theta \,\sin \phi &r\,\cos \theta \,\sin \phi &r\,\sin \theta \,\cos \phi \\\cos \theta &-r\,\sin \theta &0\end{bmatrix}}.$.

Determinanta je enaka $r^{2}\sin \theta \,$$r^{2}\sin \theta \,$.

Primer 2

Poiščimo Jacobijevo matriko za funkcijo $f:R^{3}\to R^{4}\,$$f:R^{3}\to R^{4}\,$ za komponente

$y_{1}=x_{1}\,$$y_{1}=x_{1}\,$

$y_{2}=5x_{3}\,$$y_{2}=5x_{3}\,$

$y_{3}=4x_{2}^{2}-2x_{3}\,$$y_{3}=4x_{2}^{2}-2x_{3}\,$

$y_{4}=x_{3}\sin(x_{1})\,$$y_{4}=x_{3}\sin(x_{1})\,$.

V tem primeru se dobi Jacobijeva matrika

$J_{f}(x_{1},x_{2},x_{3})={\begin{bmatrix}{\dfrac {\partial y_{1}}{\partial x_{1}}}&{\dfrac {\partial y_{1}}{\partial x_{2}}}&{\dfrac {\partial y_{1}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{2}}{\partial x_{1}}}&{\dfrac {\partial y_{2}}{\partial x_{2}}}&{\dfrac {\partial y_{2}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{3}}{\partial x_{1}}}&{\dfrac {\partial y_{3}}{\partial x_{2}}}&{\dfrac {\partial y_{3}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{4}}{\partial x_{1}}}&{\dfrac {\partial y_{4}}{\partial x_{2}}}&{\dfrac {\partial y_{4}}{\partial x_{3}}}\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&0&5\\0&8x_{2}&-2\\x_{3}\cos(x_{1})&0&\sin(x_{1})\end{bmatrix}}.$$J_{f}(x_{1},x_{2},x_{3})={\begin{bmatrix}{\dfrac {\partial y_{1}}{\partial x_{1}}}&{\dfrac {\partial y_{1}}{\partial x_{2}}}&{\dfrac {\partial y_{1}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{2}}{\partial x_{1}}}&{\dfrac {\partial y_{2}}{\partial x_{2}}}&{\dfrac {\partial y_{2}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{3}}{\partial x_{1}}}&{\dfrac {\partial y_{3}}{\partial x_{2}}}&{\dfrac {\partial y_{3}}{\partial x_{3}}}\\[3pt]{\dfrac {\partial y_{4}}{\partial x_{1}}}&{\dfrac {\partial y_{4}}{\partial x_{2}}}&{\dfrac {\partial y_{4}}{\partial x_{3}}}\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&0&5\\0&8x_{2}&-2\\x_{3}\cos(x_{1})&0&\sin(x_{1})\end{bmatrix}}.$.

Iz tega se vidi, da Jacobijeva matrika ni vedno kvadratna.

Jacobijeva determinanta

Kadar je $m=n\,$$m=n\,$ je Jacobijeva matrika kvadratna in zanjo lahko določimo determinanto. To determinanto imenujemo Jacobijeva determinanta, ki jo včasih imenujemo tudi jakobian.

Primer Jacobijeve determinante

Jacobijeva determinanta za funkcijo $f:R^{3}\to R^{3}\,$$f:R^{3}\to R^{3}\,$ s komponentami

${\begin{aligned}y_{1}&=5x_{2}\\y_{2}&=4x_{1}^{2}-2\sin(x_{2}x_{3})\\y_{3}&=x_{2}x_{3}\end{aligned}}$${\begin{aligned}y_{1}&=5x_{2}\\y_{2}&=4x_{1}^{2}-2\sin(x_{2}x_{3})\\y_{3}&=x_{2}x_{3}\end{aligned}}$

je

${\begin{vmatrix}0&5&0\\8x_{1}&-2x_{3}\cos(x_{2}x_{3})&-2x_{2}\cos(x_{2}x_{3})\\0&x_{3}&x_{2}\end{vmatrix}}=-8x_{1}\cdot {\begin{vmatrix}5&0\\x_{3}&x_{2}\end{vmatrix}}=-40x_{1}x_{2}.$${\begin{vmatrix}0&5&0\\8x_{1}&-2x_{3}\cos(x_{2}x_{3})&-2x_{2}\cos(x_{2}x_{3})\\0&x_{3}&x_{2}\end{vmatrix}}=-8x_{1}\cdot {\begin{vmatrix}5&0\\x_{3}&x_{2}\end{vmatrix}}=-40x_{1}x_{2}.$.

Glej tudi

• seznam vrst matrik

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