Matrix Analysis and Application
1 Linear Equations(线性方程组)
- Gaussian Elimination (高斯消去法)：就是用以上三种操作，选择主元，消去主元所在列以下的元素，直到变为行阶梯型。
- Gauss-Jordan Method (高斯约当法)：将主元所在列以上的部分也变为零。
- Partial Pivoting (部分主元法)：每次将当前基本列中最大元素所在方程与原主元所在方程交换位置，再用高斯消去法。
- Complete Pivoting (全部主元法)：将当前主元所在子矩阵中最大元素作为本次步骤的主元，也就是除了可以交换行，也可以交换列。交换列时需同时交换未知数。
2 Matrix Algebra(矩阵代数)
For matrices Amn and Bnm, trace(AB) = trace(BA).
trace(ABC) = trace(BCA) = trace(CAB).
trace(ABC) = trace(BCA)
trace(AB) = trace(BA)
Sherman-Morrison Formula:
2.1 LU Factorization (LU 分解)
- A = LU
- PA = LU, 系数相反数
- Existence of LU Factors
3 Vector Spaces (向量空间)
3.1 Vector Spaces and Subspaces
3.1.1 Vector Space (向量空间)
3.1.2 Subspaces (子空间)
- closure property for vector addition
- closure property for scalar multiplication
3.2 Four Fundamental Subspaces (四个基本子空间)
3.2.1 Range Spaces (值域空间)
3.2.2 Nullspace (零空间)
3.3 Linear Independence (线性独立)
3.4 Basis and Dimension
4 Linear Transformations(线性变换)
4.1 Introduction
- Linear operator (线性算子)
4.2 Invariant Subspaces(不变子空间)
5 Norms and Inner Products (模和内积)
5.1 Vector Norms
5.2 Matrix Norms
- Frobenius-norm,
- 1-norm, the largest absolute column sum
- 2-norm
- ∞-norm, the largest absolute row sum
5.3 Orthogonal Vectors (正交向量)
- Orthonormal Sets (标准正交基)
- Fourier Expansions (傅里叶展开)
5.4 Gram-Schmidt Procedure (施密特)
- Gram-Schmidt Procedure (施密特)
- Modified Gram-Schmidt Algorithm (施密特)