Lx=d: SundayResearch

Linear Algebra, 2.5, Problem 7 (c) In Gilbert Strang's Introduction to Linear Algebra 4th ed., chapter 2, section 5, Problem 7 (c) is as following. If A has row 1 + row 2 = row 3, show that A is not invertible: (c) What happens to row 3 in elimination? The question is ''what happens in elimination?'' Therefore, first I tried the elimination. It turned out it is complicated. I was in Saarbruecken last week. I have no one to meet one afternoon there, so, I compute this on a piece of paper with a pen. I sat in a cafe in Sankt Johanner Markt for two hours for this, but, I had no luck at that day. Now I had the answer, but, this is a bad answer. This is not a wrong answer, but I would say this is not a good answer. Row 3 is always (0 0 0). Actually I used a computational software. Unfortunately, this result doesn't give me a feeling ``I understand something.'' I didn't understand, Why the row 3 becomes all zero, in this way. I could not answer ...

A by-talk I was in Saarbrueecken to perticipate my friend's Polterabend. When I took my breakfast on Satutday, I met an old friend, Tom, totally unexpected. Since he now lives in US. He just visited a week here. I just talked about an article I am going to write (this one), then it turns out that he is also reading Gilbert Strang's Introduction to Linear Algebra. What a coincidence. That reminds me that the end of last year. I was in Yokohama and met an old friend from Germany. The meeting is not so coincident since there was a big event related with my job and he also worked in the area. But again, he was also reading the same book.  Murakami wrote some about Jazz's coincidence in his 'Tokyo Kitansyu (Strange stories in Tokyo)'. It is really such things happens sometimes. Appendix 1 Pairs of AB=-BA matrix.

Visualizing AB=-BA I looked into the results and found out there are many dense matrices. These are not so easy to comprehend for me. For example, I have no intuitive understanding on   Dense matrix example with a glance. I am a lazy Sunday researcher, so let's use matlab again. matlab has a eigshow, a great example code for matrix visualization. You can find the source code of that in the matlab installed directory. I read the eigshow code, then wrote a AB=-BA visualizer. The following Figures are the snapshots of the program. You will find the source code in the Appendix of this follow up blog, so you could try it if you are interested in.

Marco's Question In the conference, I met my old friend, Marco. This conference is the state of the art computer graphics conference, so, the story of 2 by 2 matrix is totally not be interedted in. However, he asked me what I am on, then I answer the matrix story. Then he asked me a question, ``Can we visualize such matrices?'' It is a brilliant question. What is the common property of these matrix? Is that any common geometrical property, for instance? The geometric property of my first answer, Equation(1) of Skewed Commutative (?) Matrix (1), is 90 degree rotation (A) and flip y direction (B). In Figure AB_BA, these matrices applied to the vector (1,0) which is shown as the red arrow. Figure AB_BA As shown in Figure AB_BA, when AB is applied to the vector (1 0)^T, first B reflects Y axis direction, but the vector (1 0)^T has no change since Y component is 0. Then A rotates the vector by 90 degrees. The result is a Y axis + direction vector, (0 1)^T. Instead, BA fi...

Norrkoeping This early of May, I visited to Norrkoeping in Sweden to perticipate a conference. This city is fantastic. Although it is neither a big city nor having a famous sight seeing point, the people living there are just nice. I arrived at there very late, around 1 am, yet it is planned, so I took a taxi to go to my hostel. The taxi driver found the address, but we could only see a train station, not see the hostel. Then the driver stopped the meter and looked for the hostel. In this case, the driver usually doesn't stop the meter, I was already impressed. Later we know the hostel was there, the 1st floor of the train station was actually the hostel. We found a small sign, that tells hostel, so I thanked him and he left. But, now the door of the hostel is closed and I found out there is no door opened. I found a sign that it seems the opening time, but it is in Swedish except numbers. If it is the opening time, I need to wait until 5 o'clock in the morning. I've ...

I worked on image processing a bit, but is this the usual people expecting? Let's enhance Reflection...

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...