Matrix Multiplication
There are 5 common multiply mmethods
1 just the define
`A = matrix([
[ ],
[ ]
[a31 a32 a33]])`
`B = matrix([
[ , b12, ],
[ , b22, ],
[ , b32, ]])
`
`C = A * B = matrix([
[ ],
[ ],
[ , c32, ]])
`
that is, c32 = a31b12 + a32b22 + a33*b32 = ∑a{3k}b{k2}; The entry in row i and column j of AB is (row i of A) · (column j of B).
2 column vector
`A = matrix([
[1,1],
[2,-1]
])
B = matrix([
[2, 2]
[3, 4]
])
C*1 = 2 * [1, 2].T + 3 [1,-1].T
C*2 = 2 * [1, 2].T + 4 [1, -1].T
`
A is column vector, and B is coefficient matrix(x, y).
3 row vector
`A = matrix([
[1,1],
[2,-1]
])
B = matrix([
[2, 2]
[3, 4]
])
R1* = 1 * [2, 2] + 1 [3,4]
R2* = 2 * [2, 2] + (-1)[3,4]
`
Just like method 2, B is row vector, and A is coefficient matrix(x, y).
4 the define apply
`A = matrix([
[1,1],
[2,-1]
])
B = matrix([
[2, 2]
[3, 4]
])
C1 = [1, 2].T * [2, 2] = [[2, 2], [4, 4]]
C2 = [1, -1].T * [3,4] = [[3, 4], [-3, -4]]
C = C1 + C2 = [[5,6],[1,0]]
`
As you can see, C = C1R1 + C2R2.
5 Block Multiplication
The Laws for Matrix Operations
Here are three addition laws:
> A+B=B+A (commutative law)
> c(A + B) = cA + cB (distributive law)
> A + (B + C) = (A + B) + C ( associative law). *** Three more laws hold for multiplication, but AB = BA is not one of them
> AB ^= BA (the commutative "law" is usually broken
> C(A + B) = CA + CB (distributive law from the left
> (A + B)C = AC + BC (distributive law from the right)
> A(BC) = (AB)C (associative law for ABC) (parentheses not needed).
Inverse
(AB)-1 = B-1A-1
There’s no inverse if some combination of the columns gives nothing.
Ax=0
| A | = 0 |
Calculating A-I by Gauss-Jordan Elimination
pivot
E-121E-131E-132 = L
LU
fuck sup sub is out of order!!!
