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.
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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