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文章目錄
- 矩陣微積分@Matrix calculus
- 記法
- 簡單Jacobi Matrix
- 分子記法
- 分母記法
- 一般形式的Jacobi Matrix
- Types of matrix derivative
- 向量求導(dǎo)
- 向量對標(biāo)量求導(dǎo)
- 標(biāo)量對向量求導(dǎo)
- 向量對向量求導(dǎo)
- 矩陣求導(dǎo)
- 矩陣對標(biāo)量求導(dǎo)(切矩陣)
- 標(biāo)量對矩陣求導(dǎo)
- 記法
- 向量求導(dǎo)
- 向量對標(biāo)量求導(dǎo)
- 標(biāo)量對向量求導(dǎo)
- 向量對向量求導(dǎo)
- 矩陣求導(dǎo)
- 矩陣對標(biāo)量求導(dǎo)
- 標(biāo)量對矩陣求導(dǎo)
矩陣微積分@Matrix calculus
- 深度學(xué)習(xí)中的矩陣微積分學(xué) - Dezeming Family https://dezeming.top ? uploads ? 2022/02 ? 深度…
- 矩陣論 第2版_圖書搜索 (superlib.net)
- **作者:**方保镕，周繼東，李醫(yī)民編著 **頁數(shù):**401 **出版社:**北京：清華大學(xué)出版社 **出版日期:**2013.12
- 簡介:本書比較全面、系統(tǒng)地介紹了矩陣的基本理論、方法及其應(yīng)用。
- 全書分上、下兩篇，上篇為基礎(chǔ)篇，下篇為應(yīng)用篇。
- Matrix calculus - Wikipedia
- 矩陣微積分 (wikipedia.org)
記法
- 在表示向量和矩陣時，通過用單個變量(字母)來表示許多變量的方式，把矩陣記法的效用發(fā)揮到最大。
- 可以用不同字體來區(qū)分標(biāo)量、向量和矩陣。
- 我們使用M(n,m)M(n,m)M(n,m)來表示包含n行m列的n×mn×mn×m實(shí)矩陣的空間。
- 該空間中的一般矩陣用大寫字母表示，例如A，X，Y等。
- 而若該矩陣屬于M(n,1)M(n,1)M(n,1)，即列向量，則用粗體小寫字母表示，如a，x，y等。(但有時為例放便,不以粗體書寫)
- 特別地，M(1,1)中的元素為標(biāo)量，用小寫斜體字母表示，如a，t，x等。
- XTX^TXT 表示矩陣轉(zhuǎn)置，tr(X)tr(X)tr(X)表示矩陣的跡，而 det?(X)\det(X)det(X)或∣X∣|X|∣X∣表示行列式。
- 除非專門注明，所有函數(shù)都默認(rèn)屬于光滑函數(shù)。
- 通常字母表前半部分的字母(a,b,c,…)(a, b, c, …)(a,b,c,…)用于表示常量
- 而后半部分的字母(t,x,y,…)(t, x, y, …)(t,x,y,…)用于表示變量。
- 變量可以是標(biāo)量,也可以是向量
- 標(biāo)量可能是常數(shù),也可能是變量
簡單Jacobi Matrix
分子記法
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這個矩陣我們稱為雅克比矩陣 (Jacobian matrix),一下是分子記法(分子布局 (numerator layout))
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J=[?f(x,y)?g(x,y)]=[?f(x,y)?x?f(x,y)?y?g(x,y)?x?g(x,y)?y]\mathcal{J}=\left[\begin{array}{l} \nabla f(x, y) \\ \nabla g(x, y) \end{array}\right]=\left[\begin{array}{ll} \frac{\partial f(x, y)}{\partial x} & \frac{\partial f(x, y)}{\partial y} \\ \frac{\partial g(x, y)}{\partial x} & \frac{\partial g(x, y)}{\partial y} \end{array}\right] J=[?f(x,y)?g(x,y)?]=[?x?f(x,y)??x?g(x,y)???y?f(x,y)??y?g(x,y)??]
分母記法
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有許多著作和軟件會使用分母布局 (denominator layout)，其實(shí)這就是分子布局的矩陣轉(zhuǎn)置：
- [?f(x,y)?x?f(x,y)?y?g(x,y)?x?g(x,y)?y]T=[?f(x,y)?x?g(x,y)?x?f(x,y)?y?g(x,y)?y]\left[\begin{array}{ll} \frac{\partial f(x, y)}{\partial x} & \frac{\partial f(x, y)}{\partial y} \\ \frac{\partial g(x, y)}{\partial x} & \frac{\partial g(x, y)}{\partial y} \end{array}\right]^{T}=\left[\begin{array}{ll} \frac{\partial f(x, y)}{\partial x} & \frac{\partial g(x, y)}{\partial x} \\ \frac{\partial f(x, y)}{\partial y} & \frac{\partial g(x, y)}{\partial y} \end{array}\right] [?x?f(x,y)??x?g(x,y)???y?f(x,y)??y?g(x,y)??]T=[?x?f(x,y)??y?f(x,y)???x?g(x,y)??y?g(x,y)??]
一般形式的Jacobi Matrix
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對于多個標(biāo)量函數(shù),講它們組合到一個向量中:
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令y=f(x)\mathbf{y}=\mathbf{f(x)}y=f(x)是一個由若干(設(shè)為m個)多元(設(shè)為n元)標(biāo)量函數(shù)構(gòu)成的向量
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把n維向量x\mathbf{x}x作為輸入,fi(x)f_i(\mathbf{x})fi?(x)返回一個標(biāo)量值(Rn→RR^n\to{R}Rn→R)
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y=(y1y2?ym)=f(x)=(f1(x)f2(x)?fm(x))\mathbf{y}=\begin{pmatrix} y_{1}\\ y_{2}\\ \vdots\\ y_{m}\\ \end{pmatrix} =\mathbf{f(x)} =\begin{pmatrix} f_{1}(\mathbf{x})\\ f_{2}(\mathbf{x})\\ \vdots\\ f_{m}(\mathbf{x})\\ \end{pmatrix} y=?y1?y2??ym???=f(x)=?f1?(x)f2?(x)?fm?(x)??
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m=nm=nm=n的情況是很常見的
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Jacobi矩陣就是與x\mathbf{x}x函數(shù)相關(guān)的的m個梯度
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?y?x=(?f1(x)?f2(x)??fm(x))=(??xf1(x)??xf2(x)???xfm(x))=(??x1f1(x)??x2f1(x)???xnf1(x)??x1f2(x)??x2f2(x)???xnf2(x)???x1fm(x)??x2fm(x)???xnfm(x))m×nX∈Rn?fi(x)=??xfi(x)=[??x1fi(x),??x2fi(x),?,??xnfi(x)]\frac{\partial\mathbf{y}}{\partial{\mathbf{x}}} =\begin{pmatrix} \nabla f_{1}{(\mathbf{x})}\\ \nabla f_{2}{(\mathbf{x})}\\ \vdots\\ \nabla f_{m}{(\mathbf{x})}\\ \end{pmatrix} =\begin{pmatrix} \frac{\partial}{\partial{\mathbf{x}}}f_{1}(\mathbf{x})\\ \frac{\partial}{\partial{\mathbf{x}}}f_{2}(\mathbf{x})\\ \vdots\\ \frac{\partial}{\partial{\mathbf{x}}}f_{m}(\mathbf{x})\\ \end{pmatrix}\\ =\begin{pmatrix} \frac{\partial}{\partial{{x_1}}}f_{1}(\mathbf{x})& \frac{\partial}{\partial{{x_2}}}f_{1}(\mathbf{x})& \cdots& \frac{\partial}{\partial{{x_n}}}f_{1}(\mathbf{x})\\ \frac{\partial}{\partial{{x_1}}}f_{2}(\mathbf{x})& \frac{\partial}{\partial{{x_2}}}f_{2}(\mathbf{x})& \cdots& \frac{\partial}{\partial{{x_n}}}f_{2}(\mathbf{x})\\ \vdots\\ \frac{\partial}{\partial{{x_1}}}f_{m}(\mathbf{x})& \frac{\partial}{\partial{{x_2}}}f_{m}(\mathbf{x})& \cdots& \frac{\partial}{\partial{{x_n}}}f_{m}(\mathbf{x})\\ \end{pmatrix}_{m\times{n}} \\ \mathbf{X}\in{R^n} \\ \nabla f_{i}{(\mathbf{x})}=\frac{\partial}{\partial{\mathbf{x}}}f_{i}(\mathbf{x}) = [\frac{\partial}{\partial{{x_1}}}f_{i}(\mathbf{x}), \frac{\partial}{\partial{{x_2}}}f_{i}(\mathbf{x}), \cdots, \frac{\partial}{\partial{{x_n}}}f_{i}(\mathbf{x})] ?x?y?=??f1?(x)?f2?(x)??fm?(x)??=??x??f1?(x)?x??f2?(x)??x??fm?(x)??=??x1???f1?(x)?x1???f2?(x)??x1???fm?(x)??x2???f1?(x)?x2???f2?(x)?x2???fm?(x)??????xn???f1?(x)?xn???f2?(x)?xn???fm?(x)??m×n?X∈Rn?fi?(x)=?x??fi?(x)=[?x1???fi?(x),?x2???fi?(x),?,?xn???fi?(x)]
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對于 fi(x)=fi([x1,x2,?,xn])=xif_i(\mathbf{x})=f_i([x_1,x_2,\cdots,x_n])= x_ifi?(x)=fi?([x1?,x2?,?,xn?])=xi? (構(gòu)成的)的恒等函數(shù) f(x)=x\mathbf{f(x)} = \mathbf{x}f(x)=x，我們可以計算得到它的雅克比矩陣（這里的 m 等于 n）
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y=f(x)=(f1(x),f2(x),?,fn(x))=x\mathbf{y=f(x)}=(f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x}))=\mathbf{x} y=f(x)=(f1?(x),f2?(x),?,fn?(x))=x
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注意這里的函數(shù)f\mathbf{f}f是向量輸入x\mathbf{x}x,同時向量輸出y\mathbf{y}y
- 假設(shè)它么的維數(shù)分別是n,mn,mn,m,且有m=nm=nm=n
- 對上述恒等函數(shù)求jacobi matrix
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?y?x=(??x1f1(x)??x2f1(x)???xnf1(x)??x1f2(x)??x2f2(x)???xnf2(x)???x1fm(x)??x2fm(x)???xnfm(x))m×n=(??x1x1??x2x1???xnx1??x1x2??x2x2???xnx2???x1xm??x2xm???xnxm)m×n=n2=(10?001?0????00?1)n×n\frac{\partial\mathbf{y}}{\partial{\mathbf{x}}} =\begin{pmatrix} \frac{\partial}{\partial{{x_1}}}f_{1}(\mathbf{x})& \frac{\partial}{\partial{{x_2}}}f_{1}(\mathbf{x})& \cdots& \frac{\partial}{\partial{{x_n}}}f_{1}(\mathbf{x})\\ \frac{\partial}{\partial{{x_1}}}f_{2}(\mathbf{x})& \frac{\partial}{\partial{{x_2}}}f_{2}(\mathbf{x})& \cdots& \frac{\partial}{\partial{{x_n}}}f_{2}(\mathbf{x})\\ \vdots\\ \frac{\partial}{\partial{{x_1}}}f_{m}(\mathbf{x})& \frac{\partial}{\partial{{x_2}}}f_{m}(\mathbf{x})& \cdots& \frac{\partial}{\partial{{x_n}}}f_{m}(\mathbf{x})\\ \end{pmatrix}_{m\times{n}} \\ =\begin{pmatrix} \frac{\partial}{\partial{{x_1}}}x_1& \frac{\partial}{\partial{{x_2}}}x_1& \cdots& \frac{\partial}{\partial{{x_n}}}x_1\\ \frac{\partial}{\partial{{x_1}}}x_2& \frac{\partial}{\partial{{x_2}}}x_2& \cdots& \frac{\partial}{\partial{{x_n}}}x_2\\ \vdots\\ \frac{\partial}{\partial{{x_1}}}x_m& \frac{\partial}{\partial{{x_2}}}x_m& \cdots& \frac{\partial}{\partial{{x_n}}}x_m\\ \end{pmatrix}_{m\times{n}=n^2} =\begin{pmatrix} 1 &0 &\cdots&0 \\ 0 &1 &\cdots&0 \\ \vdots&\vdots&\ddots&\vdots\\ 0 &0 &\cdots&1 \\ \end{pmatrix}_{n\times{n}} ?x?y?=??x1???f1?(x)?x1???f2?(x)??x1???fm?(x)??x2???f1?(x)?x2???f2?(x)?x2???fm?(x)??????xn???f1?(x)?xn???f2?(x)?xn???fm?(x)??m×n?=??x1???x1??x1???x2???x1???xm???x2???x1??x2???x2??x2???xm???????xn???x1??xn???x2??xn???xm???m×n=n2?=?10?0?01?0??????00?1??n×n?
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Types of matrix derivative
| Types | Scalar | Vector | Matrix |
|---|---|---|---|
| Scalar | ?y?x\frac{\partial y}{\partial x}?x?y? | ?y?x\frac{\partial \mathbf{y}}{\partial x}?x?y? | ?Y?x\frac{\partial \mathbf{Y}}{\partial x}?x?Y? |
| vector | ?y?x\frac{\partial y}{\partial \mathbf{x}}?x?y? | ?y?x\frac{\partial \mathbf{y}}{\partial \mathbf{x}}?x?y? | |
| Matrix | ?y?X\frac{\partial y}{\partial \mathbf{X}}?X?y? |
向量求導(dǎo)
向量對標(biāo)量求導(dǎo)
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由于向量可看成僅有一列的矩陣，最簡單的矩陣求導(dǎo)為向量求導(dǎo)。
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通過如下方式表達(dá)大部分向量微積分：
- 把n維向量構(gòu)成的空間M(n,1)等同為歐氏空間 RnR^nRn， 標(biāo)量M(1,1)M(1,1)M(1,1)等同于R。
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向量y=[y1y2?ym]T{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}y=[y1??y2????ym??]T關(guān)于標(biāo)量 xxx的導(dǎo)數(shù)可以（用分子記法）寫成
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yi=yi(x)y_i=y_i(\mathbf{x})yi?=yi?(x)多元(輸入)標(biāo)量(輸出)函數(shù)
- i=1,2,?,mi=1,2,\cdots,mi=1,2,?,m
- 對標(biāo)量xxx進(jìn)行廣播,再分別求導(dǎo)
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?y?x=[?y1?x?y2?x??ym?x]{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x}}\\{\frac {\partial y_{2}}{\partial x}}\\\vdots \\{\frac {\partial y_{m}}{\partial x}}\\\end{bmatrix}}} ?x?y?=??x?y1???x?y2????x?ym????
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在向量微積分中，向量y\mathbf {y}y關(guān)于標(biāo)量變量xxx的導(dǎo)數(shù)也被稱為向量y\mathbf {y}y的切向量(在xxx方向的)，?y?x{\displaystyle {\frac {\partial \mathbf {y} }{\partial x}}}?x?y?
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例子
- 簡單的樣例包括歐式空間中的速度向量，它是位移向量（看作關(guān)于時間的函數(shù)）的切向量。
- 更進(jìn)一步而言， 加速度是速度的切向量。
標(biāo)量對向量求導(dǎo)
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標(biāo)量y對向量x=[x1x2?xn]T{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}x=[x1??x2????xn??]T的導(dǎo)數(shù)可以（用分子記法）寫成
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?y?x=[?y?x1?y?x2??y?xn]{\displaystyle {\frac {\partial y}{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{1}}}&{\frac {\partial y}{\partial x_{2}}}&\cdots &{\frac {\partial y}{\partial x_{n}}}\end{bmatrix}}}?x?y?=[?x1??y???x2??y?????xn??y??]
- 對被求導(dǎo)的多元函數(shù)y=y(x)y=y(\mathbf{x})y=y(x)進(jìn)行廣播(broadcasting),再進(jìn)行求導(dǎo)
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(?y?x)T=(?y?x1?y?x2?y?x3)(\frac{\partial{y}}{\partial{\mathbf{x}}})^T =\begin{pmatrix} \frac {\partial y}{\partial \mathbf {x_1} }\\ \frac {\partial y}{\partial \mathbf {x_2} }\\ \frac {\partial y}{\partial \mathbf {x_3}} \end{pmatrix} (?x?y?)T=??x1??y??x2??y??x3??y???
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在向量微積分中，標(biāo)量y在的空間RnR^nRn(其獨(dú)立坐標(biāo)是x的分量)中的梯度是標(biāo)量y對向量x的導(dǎo)數(shù)的轉(zhuǎn)置。
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在物理學(xué)中，電場是電勢的負(fù)梯度向量。
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標(biāo)量函數(shù)f(x)對空間向量x在單位向量u（在這里表示為列向量）方向上的方向?qū)?shù)可以用梯度定義：
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?uf(x)=?f(x)?u\displaystyle \nabla _{\mathbf {u} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {u}?u?f(x)=?f(x)?u
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u=(cos?α,cos?β,cos?γ)u=(\cos\alpha,\cos\beta,\cos\gamma)u=(cosα,cosβ,cosγ),即該單位向量是由u方向的方向余弦構(gòu)成的
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?f(x)u=(?y?x1,?y?x2,?y?x3)?(cos?α,cos?β,cos?γ)=(?y?x1?y?x2?y?x3)(cos?α,cos?β,cos?γ)=(?y?x)Tu\nabla{f(x)}u=(\frac {\partial y}{\partial \mathbf {x_1} },\frac {\partial y}{\partial \mathbf {x_2} },\frac {\partial y}{\partial \mathbf {x_3} }) \cdot(\cos\alpha,\cos\beta,\cos\gamma) \\=\begin{pmatrix} \frac {\partial y}{\partial \mathbf {x_1} }\\ \frac {\partial y}{\partial \mathbf {x_2} }\\ \frac {\partial y}{\partial \mathbf {x_3}} \end{pmatrix}(\cos\alpha,\cos\beta,\cos\gamma) =(\frac{\partial{y}}{\partial{\mathbf{x}}})^T\mathbf{u} ?f(x)u=(?x1??y?,?x2??y?,?x3??y?)?(cosα,cosβ,cosγ)=??x1??y??x2??y??x3??y???(cosα,cosβ,cosγ)=(?x?y?)Tu
- x=(x1,x2,x3)\mathbf x=(x_1,x_2,x_3)x=(x1?,x2?,x3?)
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使用剛才定義的標(biāo)量對向量的導(dǎo)數(shù)的記法，可以把方向?qū)?shù)寫作 ?uf=(?f?x)?u\displaystyle \nabla _{\mathbf {u} }f=\left({\frac {\partial f}{\partial \mathbf {x} }}\right)^{\top }\mathbf {u}?u?f=(?x?f?)?u 這類記法在證明乘法法則和鏈?zhǔn)椒▌t的時候非常直觀，因?yàn)樗鼈兣c我們熟悉的標(biāo)量導(dǎo)數(shù)的形式較為相似。
向量對向量求導(dǎo)
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前面兩種情況可以看作是向量對向量求導(dǎo)在其中一個是一維向量情況下的特例。
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類似地我們將會發(fā)現(xiàn)有關(guān)矩陣的求導(dǎo)可被以一種類似的方式化歸為向量求導(dǎo)。
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分量為函數(shù)的向量 y=[y1y2?ym]T{\displaystyle \mathbf {y} ={\begin{bmatrix}y_{1}&y_{2}&\cdots &y_{m}\end{bmatrix}}^{\mathsf {T}}}y=[y1??y2????ym??]T對輸入向量x=[x1x2?xn]T{\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\cdots &x_{n}\end{bmatrix}}^{\mathsf {T}}}x=[x1??x2????xn??]T的導(dǎo)數(shù)x→yi(x)\mathbf{x}\to{y_i(\mathbf{x})}x→yi?(x)，可以（用分子記法) 寫作
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Note:yi=yi(x)y_i=y_i(\mathbf{x})yi?=yi?(x),i=1,2,?,mi=1,2,\cdots,mi=1,2,?,m
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這部分在開頭做過展示(Jacobi Matrix)
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?y?x=[?y1?x1?y1?x2??y1?xn?y2?x1?y2?x2??y2?xn?????ym?x1?ym?x2??ym?xn]{\displaystyle {\frac {\partial \mathbf {y} }{\partial \mathbf {x} }}={\begin{bmatrix}{\frac {\partial y_{1}}{\partial x_{1}}}&{\frac {\partial y_{1}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{1}}{\partial x_{n}}}\\{\frac {\partial y_{2}}{\partial x_{1}}}&{\frac {\partial y_{2}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{2}}{\partial x_{n}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m}}{\partial x_{1}}}&{\frac {\partial y_{m}}{\partial x_{2}}}&\cdots &{\frac {\partial y_{m}}{\partial x_{n}}}\\\end{bmatrix}}} ?x?y?=??x1??y1???x1??y2????x1??ym????x2??y1???x2??y2????x2??ym?????????xn??y1???xn??y2????xn??ym????
- 每一行相當(dāng)于函數(shù)yiy_iyi?對向量x\mathbf{x}x求導(dǎo)
- y\mathbf{y}y中包含了n個向量,所以y\mathbf{y}y對x\mathbf{x}x會產(chǎn)生n行,它們構(gòu)成矩陣?y?x\frac{\partial{\mathbf{y}}}{\partial{\mathbf{x}}}?x?y?
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向量函數(shù)
- 向量值函數(shù)，有時也稱為向量函數(shù)，是一個單變量或多變量的、值域是多維向量或者無窮維向量的集合的函數(shù)。向量值函數(shù)的輸入可以是一個標(biāo)量或者一個向量（定義域的維度可以是1或大于1）；定義域的維度不取決于值域的維度。
- A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors.
- The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1);
- the dimension of the function’s domain has no relation to the dimension of its range.
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在向量微積分中，向量函數(shù)y對分量表示一個空間的向量x\mathbf{x}x的導(dǎo)數(shù)也被稱為前推 (微分)，或雅可比矩陣。
- In vector calculus, the derivative of a vector function y with respect to a vector x whose components(分量) represent a space is known as the pushforward (or differential), or the Jacobian matrix.
- 在向量微積分中，向量函數(shù)y關(guān)于向量x的導(dǎo)數(shù)(其分量表示空間)被稱為推進(jìn)(或微分)，或稱為雅克比矩陣。
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向量函數(shù)f\mathbf{f}f對RnR^nRn空間中向量v的前推為df(v)=?f?vdv\displaystyle d\,\mathbf {f} (\mathbf {v} )={\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}d\,\mathbf {v}df(v)=?v?f?dv
矩陣求導(dǎo)
- 有兩種類型的矩陣求導(dǎo)可以被寫成相同大小的矩陣：矩陣對標(biāo)量求導(dǎo)和標(biāo)量對矩陣求導(dǎo)。
- 它們在解決應(yīng)用數(shù)學(xué)的許多領(lǐng)域常見的最小化問題中十分有用。
- 類比于向量求導(dǎo)，相應(yīng)的概念有切矩陣和梯度矩陣。
矩陣對標(biāo)量求導(dǎo)(切矩陣)
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矩陣函數(shù)Y對標(biāo)量x的導(dǎo)數(shù)被稱為切矩陣，(用分子記法)可寫成：
- ?Y?x=[?y11?x?y12?x??y1n?x?y21?x?y22?x??y2n?x?????ym1?x?ym2?x??ymn?x]{\displaystyle {\frac {\partial \mathbf {Y} }{\partial x}}={\begin{bmatrix}{\frac {\partial y_{11}}{\partial x}}&{\frac {\partial y_{12}}{\partial x}}&\cdots &{\frac {\partial y_{1n}}{\partial x}}\\{\frac {\partial y_{21}}{\partial x}}&{\frac {\partial y_{22}}{\partial x}}&\cdots &{\frac {\partial y_{2n}}{\partial x}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y_{m1}}{\partial x}}&{\frac {\partial y_{m2}}{\partial x}}&\cdots &{\frac {\partial y_{mn}}{\partial x}}\\\end{bmatrix}}} ?x?Y?=??x?y11???x?y21????x?ym1????x?y12???x?y22????x?ym2?????????x?y1n???x?y2n????x?ymn????
標(biāo)量對矩陣求導(dǎo)
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定義在元素是獨(dú)立變量的p×qp×qp×q矩陣X∈RnX\in{\mathbb{R}^n}X∈Rn上的標(biāo)量函數(shù)yyy對XXX的導(dǎo)數(shù)可以(用分子記法)寫作
- ?y?X=[?y?x11?y?x21??y?xp1?y?x12?y?x22??y?xp2?????y?x1q?y?x2q??y?xpq]{\displaystyle {\frac {\partial y}{\partial \mathbf {X} }}={\begin{bmatrix}{\frac {\partial y}{\partial x_{11}}}&{\frac {\partial y}{\partial x_{21}}}&\cdots &{\frac {\partial y}{\partial x_{p1}}}\\{\frac {\partial y}{\partial x_{12}}}&{\frac {\partial y}{\partial x_{22}}}&\cdots &{\frac {\partial y}{\partial x_{p2}}}\\\vdots &\vdots &\ddots &\vdots \\{\frac {\partial y}{\partial x_{1q}}}&{\frac {\partial y}{\partial x_{2q}}}&\cdots &{\frac {\partial y}{\partial x_{pq}}}\\\end{bmatrix}}} ?X?y?=??x11??y??x12??y???x1q??y???x21??y??x22??y???x2q??y????????xp1??y??xp2??y???xpq??y???
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定義矩陣上的重要的標(biāo)量函數(shù)包括矩陣的跡和行列式
- y(X)=∣X∣y(X)=|X|y(X)=∣X∣
- y(X)=Tr(X)y(X)=Tr(X)y(X)=Tr(X)
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類比于向量微積分，這個導(dǎo)數(shù)常被寫成如下形式：
- ?Xy(X)=?y(X)?X\displaystyle \nabla _{\mathbf {X} }y(\mathbf {X} )={\frac {\partial y(\mathbf {X} )}{\partial \mathbf {X} }} ?X?y(X)=?X?y(X)?
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類似地，標(biāo)量函數(shù)f(X)f(X)f(X)關(guān)于矩陣X在方向Y的方向?qū)?shù)可寫成
- ?Yf=tr?(?f?XY)\displaystyle \nabla _{\mathbf {Y} }f=\operatorname {tr} \left({\frac {\partial f}{\partial \mathbf {X} }}\mathbf {Y} \right) ?Y?f=tr(?X?f?Y)
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梯度矩陣經(jīng)常被應(yīng)用在估計理論的最小化問題中，比如卡爾曼濾波算法的推導(dǎo)，因此在這些領(lǐng)域中有著重要的地位。