Skip to main content

Skewed Commutative (?) Matrix (1)

Introduction to Linear Algebra by Gilbert Strang
Chapter 2, Section 4, Problem 22

Abstract

``By trial and error find real nonzero 2 by 2 matrices such that DE = -ED (not allowing DE = 0)'' is a problem 22, Chapter 2, Section 4 in Introduction to Linear Algebra 4th ed. by Gilbert Strang. When I worked on this problem, I got another question. ``How many such matrix pairs are there?'' I computed it and got an answer 56. Then I talk about this with my friend Marco, he asked me another question, ``Can we visualize them?'' Here is my answer of his question.


Linear Algebra, 2.4, Problem 22
There are many interesting questions in the book, Introduction to Linear Algebra 4th ed. by Gilbert Strang. For example, in Chapter 2, Section 4, Problem 22 is

 By trial and error find real nonzero 2 by 2 matrices such that
 A^2 = -I, BC = 0, DE = -ED (not allowing DE = 0).
By trial and error, I found the following answer.

Equation(1)

The book's solution page has a different answer with a comment, ``You can find more examples (p.521).''

Then I thought that How many such matrix pairs were there.

To to continued....
Labels:

Comments

Post a Comment

Popular posts from this blog

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix.  Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume  x  where x is in the null space of A .  The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the n...
Post a Comment
Read more

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um...
Post a Comment
Read more

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides:  A bit intuitive (for me) explanation of area of parallelogram  (to my site, external link) . 
Post a Comment
Read more