## 2009-08-28

### Cool matrix (1)

First I would like to introduce a bit of linear algebra. Then, I want to talk about a cool matrix.

Here, a matrix is a mathematical object, not the movie's matrix. It becomes a long story about the matrix, but in short, this is about linear operator.

But this is just a word substitute from ``matrix'' to ``linear operator,'' it seems I just fuzzify you. It is like saying Cat is Neko (in Japanese). Just substitute the words does not make sense for the people who don't know about cats.

If I could add a bit more words, linear operator treats addition and multiplication only. It's bit strange when I said ``linear algebra,'' that sounds something great, especially in Japanese. I don't know why. Some people misunderstood that is lofty, this sometimes annoys me, oh well. On the other hand, linear operations are combination of addition and multiplication. These are the same, but, if I said ``I study addition and multiplication'' instead of saying ```linear algebra,'' sometimes people said, ``Do you seriously study addition and multiplication at your age?''

Addition and multiplication has deep insight. I think when the mathematics thought about not only a single number (scaler), but also a vector and add vectors, this deepness was added. For instance, it is 5km from my home to my company, 10 km from my home to a film theater, and 11 km from my company to the theater. If these buildings are on a line, we can add and subtract 5, 10, 11 to find the relationship of the distance, but that is rare. In such case, we need not only the distance, but also the direction. We should compute the these distance including their direction. Mathematics could represent them by vectors. These are some kind of ``extended(*)'' numbers, but we are happy if we could do addition and multiplication on them (Since we could calculate the distance from my company to the film theater), we can not make light of these addition and multiplication. But, the bottom line is that linear algebra is addition and multiplication.

It becomes longer than I expected, so this is for today.

* There is a problem to call a vector as an extended number since it is not clear how to compare the vectors and how to divide the vectors. We can compare or divide usual numbers like 3 or 7, so if number should have these property, vector is not a number. Here we confront the problem that what is the number? But this is not today's topic, so I stop this theme here. If you want to know about some kind of numbers, I have a series of blog about Peano number. Also there are many other number systems.