## 2009-09-09

### 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    [x1Bx =   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.