NotesMITLinearAlgebra
3.乘法和逆矩阵
I.Different Views of Matrix Multiplication.
1.Defination
A* B
a11 a12 a13 a14
a21 a22 .................
.................................. *
.............................a44
A: m * n, B: n * p
Cij= SUM(Aik*Bkj), k -> [1,n];
2.matrix A * column vector collection of B.
matix* several column vector , the corresponding
3.row vector collections of A * matrix B.
severalrow vector * matrix.
4.column of A * row of B
m*1* 1*p
A*B= sum ( columns of A * rows of B )
5.BLOCK
[A11,A12] * [ B11, B12 ] = [ A11*B11 + A12*B21,A11*B12+A12*B22 ]
[A21,A22] [ B21, B22 ] [ A21*B11+ A22*B22, A21*B12+A22*B22 ]
II. INVERSE
Ifthere exists A' of A . that A' * A = I ( we call it invertible,non-singular.) , the A' is the inverse of A.
andif A is a square matrix A'*A = I = A*A';
SINGULAR case , non-invertible.
[ 1 3 ; 2 6 ];
1. the linear combination of columns of A. and all lies on (1,2) socant combine(1,0)
2. there is a lema? say that Ax = 0 and x != 0. then A has noinverse.
Intuition , that says that some/all columns of Acould form zero vector. So that they are linear dependent ? means lies onthe same direction.
Sothere will get no A' in this case.
For NON-Singular case
Howcan we get the inverse of A?
Gauss- Jordan ( solve 2 equations at once )
13
27* [ a ; b ] = [ 1 ; 0 ]
13
27 * [ c; d ] = [ 0 ; 1 ]
andhow can we solve it , use following ideas.
AI --> I A-1
look more detail for 3.乘法和逆矩阵.mp4later.