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🔠 Linear Algebra

Linear Algebra - [ 03. Multiplication and Inverse Matrices ]

date
Jun 9, 2023
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linear-algebra-03
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Public
tags
Linear Algebra
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Multiplication and Inverse Matrices
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Post
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🔠 Linear Algebra
updatedAt
Jul 28, 2023 07:29 AM
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  • 발표자는 행렬 곱셈에 대한 규칙을 논하며 다양한 방법이 모두 동일한 결과를 제공한다고 강조합니다.
  • 역행렬의 중요성과 그것을 찾는 방법에 대해 언급합니다.
  • 행렬 곱셈을 행과 열 연산을 사용하여 설명하며 특정 항목에 대한 공식도 제시합니다.
  • 발표자는 블록 단위로 행렬을 곱하는 개념을 소개하며 그 유용성을 강조합니다.
  • 가우스-조던 소거법을 소개하며 정사각 행렬의 역행렬을 찾기 위한 방법으로 언급하며 행렬을 항등 행렬로 변환하는 것을 목표로 합니다.
 

The rules for matirx multiplication

The interesting part is the many ways I can do it, and they all give the same answer. And they’re all important. Matrix multiplication and then, come inverses.
 
I’ll begin with how to multiply two matrices.
  1. Regular way
      2.  
      notion image
       
      But, I want to talk about other ways to look at that same calculation. Looking at whole columns and whole rows.
       
  1. Look at it by column
      2.  
      notion image
       
      columns of C are combinations of columns of A. ⇒ A column 의 길이도 m, C columns 길이도 n
       
  1. Look at it by row
      2.  
      notion image
       
      rows of C are combinations of rows of B.
       
  1. AB = sum of (cols of A) x (rows of B)
    1. column of A x row of B
        2.  
         
        columns of the answer : They’re all multiples of (columns of A) rows of the answer : They’re all multiples of (rows of B)
        All those rows lie on the some line through (1, 6). If I draw a picture of all these row vectors, they’re all the same direction. If I draw a picture of all these columns vectors, they’re all the same direction.
         
    2. sum of (cols of A) x (rows of B)
        3.  
         
       
  1. Block multiplication
      2. I could also cut the matrix into blocks. And do the multiplication by blocks. That’s actually so useful that I want to mention it.
       
      notion image
       
 
어쨌든 위의 5가지 방식 전부 같은 multiplications. 그리고 1번이 what we’re really doing when we do it. So, I just have to get the rules straight for matrix multiplication.
 

Inverses (square matrices)

 
 
If A inverse exists, A == invertible, non-singular.
And singular means no inverses.
 
ex)
 
I can find a Vector X ≠ 0 with A X = 0
 
 
Thoses two columns both lie on the same line. Every combination is just going to be on that line and I can’t get (1, 0)
⇒ non-invertible matrices, sigular matrices, some combinations of their columns gives the zero column. ⇒ A X = 0 (X≠0) 에서 역행렬 곱해도 X=0 에서 벗어날 수 없다. 그러므로 역행렬이 있을 수 없다.
 

Gauss - Jordan

 
 
Looking at that equation by columns → A x column j of A inverse = column j of I We’re back to solving systems of equations, but we’re solving two-right hand sides instead of one.
⇒ That’s where Jordan comes in.
 
Here’s the Gauss-Jordan idea.
Gauss-Jordan (solve 2eqns at once)
 
 
 
‘E A = I’ tells us E = A inverse.
So, ‘? = E = A inverse’
There is the statement of Gauss-Jordan elimination. That’s how to find the inverse.