Skip to main content

Skewed Commutative (?) Matrix (2)

How many such matrices are there?
I first calculated the constraint and tried to find the matrix pairs that satisfy the constraint. I.e., I compute AB and -BA,


Equation first try

then four equations I got. However, there were so many cases and I had no confidence I did this correct.

My next idea is using matlab/octave. To make the problem simple, I added the following condition, I think it is still enough interesting with this condition.

  The elements of the matrix are {-1,0,1} only.

My choise of algorithm is a simple standard 'generate and test' method, I generate 6561 (= 3^4 x 3^4) pairs of matrix, then test each matrix pair satisfying the condision AB=-BA (not allowing AB=0) or not. In this method, I generate the permulation and not the combination, therefore I counted the same pair twice. (For example, If we do D = B, E = A in Equation(1) from the last blog, AB and DE are different solutions.)

The result is 112, therefore, I found 56 pairs of matrix.
Labels:

Comments

Post a Comment

Popular posts from this blog

Geometric Multiplicity: eignvectors (2)

If eigenvectors of a matrix A are independent, it is a happy property. Because the matrix A can be diagonalized with a matrix S that column vectors are eigenvectors of A . For example, Why this is a happy property of A? Because I can find A's power easily. A^{10} is not a big deal. Because Λ is a diagonal matrix and power of a diagonal matrix is quite simple. A^{10} = SΛ^{10} S^{-1} Then, why if I want to compute power of A ? That is the same reason to find eigenvectors. Eigenvectors are a basis of a matrix. A matrix can be represented by a single scalar. I repeat this again. This is the happy point, a matrix becomes a scalar. What can be simpler than a scalar value. But, this is only possible when the matrix S's columns are independent. Because S^{-1} must be exist. Now I come back to my first question. Is the λ's multiplicity related with the number of eigenvectors? This time I found this has the name. Geometric multiplicity (GM): the number of in...
Post a Comment
Read more

Tezuka Osamu's Black Jack, "Shrinking"

I like several novel authors. My first favorite author is probably Teduka, Osamu. I still love him. The list grows by adding Hoshi, Shinichi, Agatha Christie, Hermann Hesse, and so forth. My first favorite article of Tezuka was Atom as most of the (boy's) Tezuka fans did. But my favorite is Black Jack. I try to summarize one story, it is still quite vivid in my memory. I first read this story when I was 13 - 15 years old. I re-read it at least several times since Black Jack is composed of many short episodes. The title should be "ちぢむ (SHRINKING)" or it might be "縮む(Shrinking)". (It is not so convenient to translate this to English, since English does not have a system to say the exact same word in several ways. So I just simulate it with capital letters.) Black Jack is a genius surgeon, but he does not have the license. In short, his medical activity is illegal. His skill is top level in the world, but, the fee is also out-of-law expensive. In the story ...
Post a Comment
Read more

My solution of Google drive hang up at "One moment please"

Today I installed Google drive to my Windows 7 environment to share files with my Linux machines. After sign in, the application window said "processing," then it hanged up. There was a button "you must enable javascript". I pushed it, then "One moment please..." after 5 minutes, I exited the program tried it again. It seems some security setting causes this problem. My solution: set  https://accounts.google.com  as a trusted site. Procedure: Open the control panel Go to network and control Go to Internet Options Open Security Tab Click Trusted sites Click the "site" button copy & paste  https://accounts.google.com  to "Add this website to the zone" and click Add button Now it worked for me. But if I removed this site, it still works. That puzzled me a bit...
25 comments
Read more