加法
\(\large A + B = B + A\)
\(\large (A + B) + C = A + (B + C)\)
與數相乘
\(\large (λμ)A=λ(μA)\)
\(\large (λ+μ)A =λA+μA\)
\(\large λ (A+B)=λA+λB\)
矩陣相乘
\(\large (AB)C = A(BC)\)
\(\large A(B \pm C) =AB \pm AC\)
\(\large (B \pm C)A =BA \pm CA\)
\(\large (λA)B = λ(AB) = A(λB)\)
轉置
記做 \(\large A^{T}\) 或 \(\small A^{`}\)
\(\large (A^{T})^{T} = A\)
\(\large (A+B)^{T} = A^{T} + B^{T}\)
\(\large (AB)^{T} = B^{T}A^{T}\)
\(\large (λA)^{T} = λA^{T}\)
導數
https://en.wikipedia.org/wiki/Matrix_calculus#Derivatives_with_vectors
布局
矩陣求導結果有兩種寫法
分子布局
\(\large \frac{\partial Y}{\partial x}=\begin{bmatrix}\frac{\partial y_{11}}{\partial x} &...& \frac{\partial y_{1j}}{\partial x} & ... & \frac{\partial y_{1m}}{\partial x} \\\frac{\partial y_{i1}}{\partial x} &...& \frac{\partial y_{ij}}{\partial x} & ... & \frac{\partial y_{im}}{\partial x}\\ \frac{\partial y_{n1}}{\partial x} &...& \frac{\partial y_{nj}}{\partial x}& ... &\frac{\partial y_{nm}}{\partial x} \end{bmatrix}\)
分母布局
\(\large \frac{\partial y}{\partial X}=\begin{bmatrix}\frac{\partial y}{\partial x_{11}} &...& \frac{\partial y}{\partial x_{1j}} & ... & \frac{\partial y}{\partial x_{1m}} \\\frac{\partial y}{\partial x_{i1}} &...& \frac{\partial y}{\partial x_{ij}} & ... & \frac{\partial y}{\partial x_{im}}\\ \frac{\partial y}{\partial x_{n1}} &...& \frac{\partial y}{\partial x_{nj}}& ... &\frac{\partial y}{\partial x_{nm}}\end{bmatrix}\)
