🔠 Linear Algebra
Linear Algebra - [ 03. Multiplication and Inverse Matrices ]
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Jun 9, 2023
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linear-algebra-03
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Linear Algebra
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Multiplication and Inverse Matrices
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🔠 Linear Algebra
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Jul 28, 2023 07:29 AM
- 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.
- Regular way
But, I want to talk about other ways to look at that same calculation. Looking at whole columns and whole rows.
- Look at it by column
columns of C are combinations of columns of A.
⇒ A column 의 길이도 m, C columns 길이도 n
- Look at it by row
rows of C are combinations of rows of B.
- AB = sum of (cols of A) x (rows of B)
- column of A x row of B
- sum of (cols of A) x (rows of B)
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.
- Block multiplication
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.
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.