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What Matrix is this?

  1. Jul 17, 2010 #1
    1. The problem statement, all variables and given/known data

    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
    ;


    2. Relevant 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?


    3. 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?
     
  2. jcsd
  3. Jul 17, 2010 #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
     
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