2010-02-01

Something I'd like to talk about determinant

Abstract

Determinant is a single number that includes many information about a matrix. Usually, we don't know about the property of a matrix at a glance, since it has a lot of numbers in it. But, if we know the determinant of the matrix, we could know about it in some extend. For example, determinant shows us the existence of the inverse of the matrix. Therefore, determinant of a matrix is an important concept in linear algebra and it is worth to think about it.


Overview of Determinant

Determinant is a single number (a scaler) that includes many information about a matrix. Since a matrix has a lot of numbers in it, it is usually hard to understand the property of a matrix at a glance. However, if we know the determinant of the matrix, we know about the matrix in some extend.

For example, if I want to introduce a man, I could explain his each detail, like when he wakes up usually, which train he takes to come to work, etc.. However, this is usually too much not-so-important information. It is commonly better to take a few brief interesting information instead of every detail: what is his job, he is gentle, he is rich, and so on. Although these features of person can't describe all the personality, it usually helps to grab the overview of the person. A property is more useful when it is simpler and more important.

Determinant can describe important properties of the matrix as the person's important properties. Especially it is just a single number (a scaler value). Such simplicity of determinant is great.

On the other hand, determinant can't describe everything of the matrix, sometimes it is too dry. Each matrix has features, some people have a preference on matrices. Seeing matrix itself is as reading in a historical story book about Wu Zixu, Sakamoto Ryoma, Maeda Keijirou, and Caesar. Getting determinant is as reading them in a history textbook. Historical story books light my heart, but I can't read all the people's books.

If someone asked a question ``What kind of person A was?'', then a simple and important answer gives us a great overview of the person A. For example, ``the person A became a King at the end.'' If I see a matrix and want to know the overview of it, determinant is a good index to start.

Matrix is versatile. We could represent many things by matrices. First, I would like to describe a complex buzz word --- linearity -- and matrix.