LA 学习笔记 Ch1 Matrix

1.周六把前三章复习完
2.Review的所有题！！

1.5 p32

First

Ax = b <=> Ux = c, where U - a new coefficient matrix

E F G elementary matrices: (step1, 2, 3) 有固定填充格式！

i - j*L
=> (-L) into (i, j)
G F E A = U,

E^-1 F^-1 G^-1 * U = A where (E^-1 F^-1 G^-1) = L
so LU = A. Triangular factorization
1H: Triangular factorization with no exchange of row. 没换行。

注意:LU（尤其是U）并不一定1s on the diagonal.

Second p36

A = LDU

D - diagonal matrix of pivots.

Third

Permutation!
1s on different row and column.
P^-1 is always the same as P^T

(p38注意先操作的要在左侧，且最接近被操作矩阵。)

PA = LDU

1.6 Inverse and Transposes

A*A^-1 = I

• Ax = 0时没有逆矩阵。- No matrix can bring 0 back to x!! x = A^-1 * Ax = A^-1
• 0 - 不可能的！
  【2x2系统有公式, 别忘了前面的系数1/(ad-bc)】
• 两个逆矩阵的乘积也是逆矩阵
• False. A matrix cannot necessarily be factored into the form A = LU because you may need to do row exchanges in order for elimination to succeed.
  (True or false)

性质（p46）:

1. 2 by 2 inverse: ad-bc ==0 (the determinant is not zero!).
2. Inverse of diagonal matrix - no diagonal entries are zero.
   (d_n => (1/d_n))
3. (AB)^-1 = (B^-1)*(A-1). The inverse come in reverse order.
   (ABC)^-1 = (C^-1)*(B-1)*(A^-1)

Gauss-Jordon method to calculate inverse:
[A e1 e2 e3] =======[I A^-1] (I - identity matrix)
通过1某行减去某行倍数Eij 2某行自己倍数D 3换行Pij 操作；
后面矩阵的变化，表示对A进行的操作（前乘矩阵: "every G-J step is a multiplication on the left by an elementary matrix"）
（操作时别忘了后面的e1 e2 e3...）

Notice: when there are zeros below AND above pivots, then the product of the pivots/diagonals = determinant!

• Invertible = nonsingular => determinant, pivots, all eigenvalues == 0
  (pivots == 0 所以中轴线上不能有0）

The transpose matrix: p50

Symmetrc Matrices: A^T = A

(AB)^T = (B^T)(AT)
(A^-1)T = (A^T)-1, therefore, (A^-1)T * A^T = I