2011-04-16

(1) Max determinant problem

Abstract

Gilbert Strang asked us what is the maximal determinant if the matrix has only specific numbers in his book, Introduction to Linear algebra. I enjoyed this problem for almost three weeks also as a programming problem. So I would like to introduce this problem in this article.


Introduction

My primary school has words, ``Be one day as one step of your life (一日生きることが一歩生きることであれ.)'' by Yukawa Hideki. These days I finally start to understand these words. I can only do something if I could do every day. Even for five minutes, if I do something every day, I found quite difference. Recently, I joined an activity. It took some significant time from my Sunday research time, though I would like to continue both my activity and my Sunday research.

At the end of March, I learn max determinant problem that exists. I didn't have any dedicated time for this problem. But, I use my commune time and elevator waiting time, I solved this problem. (Our company's elevator gives a lot of time, I usually use it for reading a book.) Using fraction time might be a point of continue something.

I didn't know why max determinant problem caught an interest of mathematicians until I started to solve this problem. In a Gilbert Strang's class, he said ``Determinant used to be very important for linear algebra''.  It was a past tense. I didn't recall that he mentioned why it was once important and not now anymore. If we think about a matrix as a linear operator, determinant is zero or not is important since it tells the system has a solution or not. I thought maximal value is not so important comparing to this.

The determinant of a matrix is magnification factor when we think the matrix is an operator. Why this is interesting? I could imagine that the absolute maximal value is less than one or not is interesting. We usually think about multiplication of matrix. For example, M^k v. But, in this case, eigenvalue is much interesting. Since if we could know the eigenvalues, this becomes M^ k v = λ^k v. This is much simpler and easy because a matrix becomes now one scalar value.

I can also think about another property of determinant, geometrical meaning. This is a volume of limited coordinates geometry. (Marc also noted this to me.) I like geometry, so, in the following articles, I will use this approach once. But, is it really interesting? This was a question to me.

I research why mathematicians are interested in max determinant problem a bit. I could not find the direct answer, but, I have an idea about that. So, I will tell about that in the next article.

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