Skip to main content

cool matrix (5)

Finally I would like to talk about cool matrices.

Let's think about a three dimensional vector [x1 x2 x3]' =


x1
[x2]
x3.


(Here ' means transpose.) The vector of difference of each term is [(x1 - 0) (x2 - x1) (x3 - x1)]'. Let's think about a matrix A that makes this difference in the right hand side. I will tell you why we think such a matrix. A is


[ 1 0 0
-1 1 0
0 -1 1].


Let's check it out.


[ 1 0 0 [x1 [ x1 - 0
-1 1 0 x2 = x2 - x1
0 -1 1] x3] x3 - x2 ]


For instance, use [x1 x2 x3]' = [1 4 9]',


[ 1 0 0 [1 [ 1
-1 1 0 4 = 3
0 -1 1] 9] 5 ].


The right hand side is the difference of each term. By the way, this matrix has its inverse matrix B,


[1 0 0
1 1 0
1 1 1].


Let's compute A B,


[ 1 0 0 [1 0 0 [ 1 0 0 [1 0 0
-1 1 0 1 1 0 = (-1+1) 1 0 = 0 1 0
0 -1 1] 1 1 1] (-1+1) (-1+1) 1] 0 0 1].


Therefore, A^{-1} = B. Please be patience a bit more. Let's look into what is B.


[ 1 0 0 [x1 [x1
Bx = 1 1 0 x2 = x1 + x2
1 1 1] x3] x1 + x2 + x3]


You see B is adding terms. A does difference and B does sum. I was surprised here. The idea ``difference'' belongs more to calculus, compare to linear algebra. In calculus, difference is d/dx. The inverse operation of difference is integral (\int). These matrices are:

A ... d/dx
B ... \int

and we know A^{-1} = B.

A^{-1} = {d/dx}^{-1} = B = \int
===> {d/dx}^{-1} = \int

Matrix behaves an operator, now we think matrix A is a differential operator and B is an integral operator, then these are inverse operation each other. I hope you can see why I surprised. We can see an analogy of the fundamental theorem of calculus in a matrix operation. So I think these matrix A and B are cool matrices. I might be only one to think so, I can imagine many people don't agree with me... Well.


This matrix story is in the book: Gilbert Strang, ``Introduction to LINEAR ALGEBRA''. This is my personal hot book. He describes what is the inverse of matrix with calculus analogy. That is fantastic and I like his book.

For the people who is lazy as me, here is a octave code.


--- BEGIN diff_and_int.m ---
a = [1 0 0; -1 1 0 ; 0 -1 1]
inv(a)
--- END diff_and_int.m ---


When you run this code with typing 'octave diff_and_int.m', you will see
the matrix B.

Comments

Popular posts from this blog

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix.  Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume  x  where x is in the null space of A .  The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the null spa

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides:  A bit intuitive (for me) explanation of area of parallelogram  (to my site, external link) .