矩阵的运算规则

2020-03-08 本文已影响0人 moon_light_

加法
$\small A + B = B + A$
$\small (A + B) + C = A + (B + C)$

与数相乘
$\small (λμ)A=λ(μA)$
$\small (λ+μ)A =λA+μA$
　　 $\small λ (A+B)=λA+λB$

矩阵相乘
$\small (AB)C = A(BC)$
$\small A(B \pm C) =AB \pm AC$
$\small (B \pm C)A =BA \pm CA$
$\small (λA)B = λ(AB) = A(λB)$

转置
记做 $\small A^{T}$ 或 $\small A^{`}$

$\small (A^{T})^{T} = A$
$\small (A+B)^{T} = A^{T} + B^{T}$
$\small (AB)^{T} = B^{T}A^{T}$
$\small (λA)^{T} = λA^{T}$

导数
https://en.wikipedia.org/wiki/Matrix_calculus#Derivatives_with_vectors

布局
矩阵求导结果有两种写法
分子布局

$\normalsize \frac{\partial Y}{\partial x}=\begin{bmatrix}\frac{\partial y_{11}}{\partial x} &...& \frac{\partial y_{1j}}{\partial x} & ... & \frac{\partial y_{1m}}{\partial x} \\\frac{\partial y_{i1}}{\partial x} &...& \frac{\partial y_{ij}}{\partial x} & ... & \frac{\partial y_{im}}{\partial x}\\ \frac{\partial y_{n1}}{\partial x} &...& \frac{\partial y_{nj}}{\partial x}& ... &\frac{\partial y_{nm}}{\partial x} \end{bmatrix}$

分母布局

$\normalsize \frac{\partial y}{\partial X}=\begin{bmatrix}\frac{\partial y}{\partial x_{11}} &...& \frac{\partial y}{\partial x_{1j}} & ... & \frac{\partial y}{\partial x_{1m}} \\\frac{\partial y}{\partial x_{i1}} &...& \frac{\partial y}{\partial x_{ij}} & ... & \frac{\partial y}{\partial x_{im}}\\ \frac{\partial y}{\partial x_{n1}} &...& \frac{\partial y}{\partial x_{nj}}& ... &\frac{\partial y}{\partial x_{nm}}\end{bmatrix}$

矩阵的运算规则

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