Lecture 07
2021-11-21
7.1. Elementary Matrix - 初等矩阵
- H_{m\times n}
Corresponding elementary row/col. op. : Interchange
- J_{m\times n}
Corresponding elementary row/col. op. : Add \lambda i to j
- F
Corresponding elementary row/col. op. : Multiply i by \lambda
7.2. - 相抵标准型
7.3. - 可逆矩阵分解定理
Let A \in M_{n}, then following statements are equivalent:
- A is invertible
- Exists a
7.4. Find the Invert of Elementary Matrix - 初等矩阵求逆
Gaussian Elimination
By Row Operation
If A \in M_n is an invertible matrix, then exists a array of n\times n elementary matrix \{P_1, P_2, \dots, P_k\} that
\begin{aligned}
(P_1, P_2, \dots, P_k)A = E_n
\end{aligned}
then
\begin{aligned}
(A | E_n) \Rightarrow ((P_1P_2\dots P_k)A | (P_1P_2\dots P_k)E_n) = (E_n | A')
\end{aligned}
we write (P_1P_2\dots P_k) = P
\begin{aligned}
PA &= E_n \quad P=A^{-1}\\
((P_1P_2\dots P_k)A | (P_1P_2\dots P_k)E_n) &= (PA|PE_n) = (E_n | P) = (E_n | A^{-1})
\end{aligned}
By Column Operation
If A \in M_n is an invertible matrix, then exists a array of n\times n elementary matrix \{Q_1, Q_2, \dots, Q_k\} that
\begin{aligned}
A(Q_1, Q_2, \dots, Q_k) = E_n
\end{aligned}
then
\begin{aligned}
(A | E_n) \Rightarrow (A(Q_1, Q_2, \dots, Q_k) | E_n(Q_1, Q_2, \dots, Q_k)) = (E_n | A')
\end{aligned}
we write (Q_1, Q_2, \dots, Q_k) = Q
\begin{aligned}
AQ &= E_n \quad P=Q^{-1}\\
(A(Q_1, Q_2, \dots, Q_k) | E_n(Q_1, Q_2, \dots, Q_k)) &= (AQ|E_nQ) = (E_n | Q) = (E_n | A^{-1})
\end{aligned}
The procedure cannot Gaussian Elimination cannot mix the row operation and col. operation
Take (P_kP_{k-1}\dots P_1)A(Q_1Q_2\dots Q_l) = E_n
This denotes that A = P^{-1}Q^{-1} \Rightarrow A^{-1} = QP
But from the difinition that
\begin{aligned}
&((P_kP_{k-1}\dots P_1)A(Q_1Q_2\dots Q_l) | (P_kP_{k-1}\dots P_1)E_n(Q_1Q_2\dots Q_l))\\
&\rightarrow (E_n | PQ)
\end{aligned}
QP \ne PQ
LU decomposition
For a invertiable matrix A, we can decompose A into A = LU,
where L is a unitary lower triangle matrix
L =
\begin{bmatrix}
1 &0 &\cdots &0\\
* &1 &\cdots &0\\
\vdots &\vdots &\vdots &\vdots\\
* &* &\cdots &1\\
\end{bmatrix}
and U is a upper triangle matrix
U =
\begin{bmatrix}
* &* &\cdots &*\\
0 &* &\cdots &*\\
\vdots &\vdots &\vdots &\vdots\\
0 &0 &\cdots &*\\
\end{bmatrix}
Method. 1 Gaussian Transformation
Let L = L_kL_{k-1}\dots L_1, then for (A | E_n)
Let LA = U, where A \in M_{m\times n}, L \in M_{m\times m}, U \in {M_{m\times n}}.
(A | E_n) \rightarrow (LA | LE_n ) \rightarrow (U | L^{-1})
Where the allowed row operation is only multiply
