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矩阵基础(Matrix Preliminary)
矩阵类型(Matrix Types)
矩阵运算(Matrix Operations)
矩阵微积分(Matrix Calculus)
∂ a T X b ∂ X = a b T \begin{aligned} \frac{\partial\mathbf{a^TXb}}{\partial\mathbf{X}}=\mathbf{a}\mathbf{b^T} \end{aligned} ∂X∂aTXb=abT ∂ a T X T b ∂ X = b a T \begin{aligned} \frac{\partial\mathbf{a^TX^Tb}}{\partial\mathbf{X}}=\mathbf{ba^T} \end{aligned} ∂X∂aTXTb=baT ∂ A = 0 \begin{aligned} \partial\mathbf{A}=0 \end{aligned} ∂A=0 ∂ ( X + Y ) = ∂ X + ∂ Y \begin{aligned} \partial(\mathbf{X+Y})= \partial\mathbf{X}+\partial\mathbf{Y}\end{aligned} ∂(X+Y)=∂X+∂Y ∂ ( α X ) = α ∂ X \begin{aligned} \partial(\alpha\mathbf{X})=\alpha\partial\mathbf{X} \end{aligned} ∂(αX)=α∂X ∂ ( α X ∘ Y ) = ( ∂ X ) ∘ Y + X ∘ ( ∂ Y ) \begin{aligned} \partial(\alpha\mathbf{X\circ Y})=(\partial\mathbf{X})\circ\mathbf{Y}+\mathbf{X}\circ(\partial\mathbf{Y}) \end{aligned} ∂(αX∘Y)=(∂X)∘Y+X∘(∂Y) ∂ ( X Y ) = ( ∂ X ) Y + X ( ∂ Y ) \begin{aligned} \partial(\mathbf{XY})=(\partial\mathbf{X})\mathbf{Y}+\mathbf{X}(\partial\mathbf{Y}) \end{aligned} ∂(XY)=(∂X)Y+X(∂Y) ∂ ( X ⊗ Y ) = ( ∂ X ) ⊗ Y + X ⊗ ( ∂ Y ) \begin{aligned} \partial(\mathbf{X\otimes Y})=(\partial\mathbf{X})\otimes\mathbf{Y}+\mathbf{X}\otimes(\partial\mathbf{Y}) \end{aligned} ∂(X⊗Y)=(∂X)⊗Y+X⊗(∂Y) ∂ ( det ( X ) ) = det ( X ) Tr ( X − 1 ∂ X ) \begin{aligned} \partial(\text{det}(\mathbf{X}))=\text{det}(\mathbf{X})\text{Tr}(\mathbf{X}^{-1}\partial\mathbf{X}) \end{aligned} ∂(det(X))=det(X)Tr(X−1∂X) ∂ ( ln ( det ( X ) ) ) = Tr ( X − 1 ∂ X ) \begin{aligned} \partial(\text{ln}(\text{det}(\mathbf{X})))=\text{Tr}(\mathbf{X}^{-1}\partial\mathbf{X}) \end{aligned} ∂(ln(det(X)))=Tr(X−1∂X)
Assume F ( X ) \begin{aligned} F(\mathbf{X}) \end{aligned} F(X) is an element-wise differentiable function.
f() is the scalar derivative of F().
∂ Tr ( F ( X ) ) ∂ X = f ( X ) T \begin{aligned} \frac{\partial\text{Tr}(F(\mathbf{X}))}{\partial\mathbf{X}}=f(\mathbf{X})^T \end{aligned} ∂X∂Tr(F(X))=f(X)T ∂ ∂ X Tr ( X A ) = A T \begin{aligned} \frac{\partial}{\partial\mathbf{X}}\text{Tr}(\mathbf{XA})=\mathbf{A}^T \end{aligned} ∂X∂Tr(XA)=AT ∂ ∂ X Tr ( A X B ) = A T B T \begin{aligned} \frac{\partial}{\partial\mathbf{X}}\text{Tr}(\mathbf{AXB})=\mathbf{A}^T\mathbf{B}^T \end{aligned} ∂X∂Tr(AXB)=ATBT ∂ ∂ X Tr ( A X T B ) = B A \begin{aligned} \frac{\partial}{\partial\mathbf{X}}\text{Tr}(\mathbf{AX^TB})=\mathbf{BA} \end{aligned} ∂X∂Tr(AXTB)=BA ∂ ∂ X Tr ( X T A ) = A \begin{aligned} \frac{\partial}{\partial\mathbf{X}}\text{Tr}(\mathbf{X^TA})=\mathbf{A} \end{aligned} ∂X∂Tr(XTA)=A ∂ ∂ X Tr ( X 2 ) = 2 X T \begin{aligned} \frac{\partial}{\partial\mathbf{X}}\text{Tr}(\mathbf{X^2})=2\mathbf{X^T} \end{aligned} ∂X∂Tr(X2)=2XT
矩阵分解(Matrix Decomposition/Factorization)
分解目的与分解手段
分解手段细说
应用举例01 – Eigen库
| 模块 | 头文件 | 内容 |
|---|---|---|
| Core | #include <Eigen/Core> | Matrix与Array类，基本线性代数运算以及数组运算 |
| Geometry | #include <Eigen/Geometry> | 变换、平移、缩放、旋转(2D与3D)[需要三维几何、世界坐标、相机坐标等知识] |
| LU | #include <Eigen/LU> | 求逆、行列式、LU分解等求解器 |
| Cholesky | #include <Eigen/Cholesky> | LLT以及LDLT乔里斯基/柯列斯基分解等求解器 |
| Householder | #include <Eigen/Householder> | 辅助变换 |
| SVD | #include <Eigen/SVD> | SVD分解以及最小二乘求解器 |
| QR | #include <Eigen/QR> | QR分解等求解器 |
| Eigenvalues | #include <Eigen/Eigenvalues> | 特征值与特征向量求解器 |
| Sparse | #include <Eigen/Sparse> | 稀疏矩阵存储以及基本的线性代数运算 |
| #include <Eigen/Dense> | 汇总头文件，包括Core、Geometry等头文件 | |
| #include <Eigen/Eigen> | 汇总头文件，包括针对稠密与稀疏矩阵操作的所有头文件(实际编程时仅需引用此文件) |
应用举例02 – MATLAB如何求解线性方程Ax=B
参考
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