- #1
patiobarbecue
- 10
- 0
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?