What Matrix is This? References for Structured Matrix

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In summary, the matrix A and B represent all possible selections from three groups (A=1,2; B=1,2,3; C=1,2,3) and have a visually appealing pattern when rearranging rows. However, B'B is singular and it is difficult to derive the pattern. It is not a well-known matrix.
  • #1
patiobarbecue
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Homework Statement



Are there references of the following structured matrix?

A =
1 0 1 0 0
1 0 0 1 0
1 0 0 0 1
0 1 1 0 0
0 1 0 1 0
0 1 0 0 1
;

Say we have group A=1,2; group B=1,2, 3; the rows of above matrix shows all possible selections: one element from each group. Hence A corresponding to the list:
11
12
13
21
22
23
;


B =
1 0 1 0 0 1 0 0
1 0 1 0 0 0 1 0
1 0 1 0 0 0 0 1
1 0 0 1 0 1 0 0
1 0 0 1 0 0 1 0
1 0 0 1 0 0 0 1
1 0 0 0 1 1 0 0
1 0 0 0 1 0 1 0
1 0 0 0 1 0 0 1
0 1 1 0 0 1 0 0
0 1 1 0 0 0 1 0
0 1 1 0 0 0 0 1
0 1 0 1 0 1 0 0
0 1 0 1 0 0 1 0
0 1 0 1 0 0 0 1
0 1 0 0 1 1 0 0
0 1 0 0 1 0 1 0
0 1 0 0 1 0 0 1
;

group A=1,2; B=1,2,3; C=1,2,3; and the above matrix indicating the list:
111
112
113
121
122
123
131
132
133
211
212
213
221
222
223
231
232
233
;


Homework Equations



Say the number of elements in a group and number of groups are arbitrary, then the linear system Ax=b, b not zero sometimes have solution; most time it is over determined, and a least square solution is x = (A'A)^-1A'b if (A'A) is invertible, when it is invertible?


The Attempt at a Solution



(A'A) is invertible and its inverse is visually appealing:

0.25000 0.50000 -0.25000 0.00000
-0.25000 0.00000 0.25000 0.50000
0.50000 -0.25000 0.50000 -0.25000
0.00000 0.25000 0.00000 0.25000

(B'B) is singular from Octave.

It seems that is a pattern, but it is pretty hard to derive it. Is this matrix a well known matrix?
 
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  • #2
By rearrange rows of B, I got P, and M=P'*P seems to have a nice pattern, M is singular though. Is it obvious why it is singular?

# Created by Octave 3.2.3, Sat Jul 17 18:31:46 2010 EDT
# name: M
# type: matrix
# rows: 8
# columns: 8
9 0 3 3 3 3 3 3
0 9 3 3 3 3 3 3
3 3 6 0 0 2 2 2
3 3 0 6 0 2 2 2
3 3 0 0 6 2 2 2
3 3 2 2 2 6 0 0
3 3 2 2 2 0 6 0
3 3 2 2 2 0 0 6
# name: A
# type: matrix
# rows: 18
# columns: 8
1 0 1 0 0 1 0 0
1 0 1 0 0 0 1 0
1 0 1 0 0 0 0 1
1 0 0 1 0 1 0 0
1 0 0 1 0 0 1 0
1 0 0 1 0 0 0 1
1 0 0 0 1 1 0 0
1 0 0 0 1 0 1 0
1 0 0 0 1 0 0 1
0 1 1 0 0 1 0 0
0 1 1 0 0 0 1 0
0 1 1 0 0 0 0 1
0 1 0 1 0 1 0 0
0 1 0 1 0 0 1 0
0 1 0 1 0 0 0 1
0 1 0 0 1 1 0 0
0 1 0 0 1 0 1 0
0 1 0 0 1 0 0 1
# name: P
# type: matrix
# rows: 18
# columns: 8
1 0 1 0 0 1 0 0
1 0 1 0 0 0 1 0
1 0 1 0 0 0 0 1
1 0 0 1 0 0 0 1
1 0 0 1 0 0 1 0
1 0 0 1 0 1 0 0
1 0 0 0 1 1 0 0
1 0 0 0 1 0 1 0
1 0 0 0 1 0 0 1
0 1 1 0 0 0 0 1
0 1 1 0 0 0 1 0
0 1 1 0 0 1 0 0
0 1 0 1 0 1 0 0
0 1 0 1 0 0 1 0
0 1 0 1 0 0 0 1
0 1 0 0 1 0 0 1
0 1 0 0 1 0 1 0
0 1 0 0 1 1 0 0
 

FAQ: What Matrix is This? References for Structured Matrix

What is a structured matrix?

A structured matrix is a matrix with a specific pattern or structure that can be identified and utilized to perform operations more efficiently. Examples of structured matrices include diagonal matrices, triangular matrices, and sparse matrices.

How is a structured matrix different from a regular matrix?

A structured matrix has a specific pattern or structure, while a regular matrix does not. This means that operations on a structured matrix can be performed more efficiently and with less computational resources compared to a regular matrix.

What are some common applications of structured matrices?

Structured matrices are commonly used in various fields such as engineering, physics, and computer science. Some specific applications include solving systems of linear equations, data compression, and image processing.

What are some references for learning more about structured matrices?

There are many resources available for learning about structured matrices, including textbooks, online courses, and research papers. Some recommended references include "Structured Matrices in Mathematics, Computer Science, and Engineering" by Robert B. Bapat and "Structured Matrices and Polynomials" by Victor Pan.

How can structured matrices benefit scientific research?

Structured matrices can greatly benefit scientific research by allowing for more efficient and accurate computations. This can lead to faster and more reliable results, as well as the ability to handle larger and more complex datasets. Additionally, structured matrices can also help to reduce computational costs and resources, making research more cost-effective.

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