生成矩阵的线性组合 [英] Generating linear combination of a matrix
问题描述
我要创建一个矩阵A [4x8]，如下所示.
矩阵A始终以1作为对角线. A11,A22,A33,A44 = 1
这个矩阵可以看作是两半，前半部分是前4列，后半部分是后4列，如下所示:
1 -1 -1 -1 1 0 0 1
A = -1 1 -1 0 0 1 0 0
-1 -1 1 0 1 0 0 0
-1 -1 -1 1 1 1 0 0
上半部分的每一行可以有两个或三个-1:
- 如果有两个
-1，则下半部分的相应行应具有一个1 - 如果任何行具有三个
-1，则矩阵的后半部分应具有两个1.
总体目标是使每行的总和为0.我需要生成像这样的矩阵的所有可能组合.
最好在每次迭代时创建具有新组合的矩阵，以便在使用后可以丢弃它，否则存储所有组合会占用大量空间.有谁能够帮我 ?
我想到的一种可能的解决方案是生成row1，row2，row3和row4的所有可能组合，并在每次迭代中创建一个矩阵.看起来可行吗?这里是一种可能的解决方案.如果暂时忽略对角线，则可以使用函数 KRON ， REPMAT ， PERMS ， ONES :
>> rowPatterns = [kron(eye(3)-1,ones(4,1)) ... %# For 2 out of 3 as -1
repmat(eye(4),3,1); ... %# For 1 out of 4 as 1
repmat([-1 -1 -1],6,1) ... %# For 3 out of 3 as -1
unique(perms([1 1 0 0]),'rows')] %# For 2 out of 4 as 1
rowPatterns =
0 -1 -1 1 0 0 0
0 -1 -1 0 1 0 0
0 -1 -1 0 0 1 0
0 -1 -1 0 0 0 1
-1 0 -1 1 0 0 0
-1 0 -1 0 1 0 0
-1 0 -1 0 0 1 0
-1 0 -1 0 0 0 1
-1 -1 0 1 0 0 0
-1 -1 0 0 1 0 0
-1 -1 0 0 0 1 0
-1 -1 0 0 0 0 1
-1 -1 -1 0 0 1 1
-1 -1 -1 0 1 0 1
-1 -1 -1 0 1 1 0
-1 -1 -1 1 0 0 1
-1 -1 -1 1 0 1 0
-1 -1 -1 1 1 0 0
请注意，对于任何给定的行，这是18个可能的模式，因此矩阵A可以具有18 ^ 4 = 104,976个可能的行模式(有点).您可以使用 NDGRID CAT 和A ="nofollow"> TRIL ， TRIU ， EYE 和 解决方案
Here's one possible solution. If you ignore the diagonal ones for the moment, you can generate all possible patterns for the other 7 values using the functions KRON, REPMAT, PERMS, UNIQUE, EYE, and ONES:
>> rowPatterns = [kron(eye(3)-1,ones(4,1)) ... %# For 2 out of 3 as -1
repmat(eye(4),3,1); ... %# For 1 out of 4 as 1
repmat([-1 -1 -1],6,1) ... %# For 3 out of 3 as -1
unique(perms([1 1 0 0]),'rows')] %# For 2 out of 4 as 1
rowPatterns =
0 -1 -1 1 0 0 0
0 -1 -1 0 1 0 0
0 -1 -1 0 0 1 0
0 -1 -1 0 0 0 1
-1 0 -1 1 0 0 0
-1 0 -1 0 1 0 0
-1 0 -1 0 0 1 0
-1 0 -1 0 0 0 1
-1 -1 0 1 0 0 0
-1 -1 0 0 1 0 0
-1 -1 0 0 0 1 0
-1 -1 0 0 0 0 1
-1 -1 -1 0 0 1 1
-1 -1 -1 0 1 0 1
-1 -1 -1 0 1 1 0
-1 -1 -1 1 0 0 1
-1 -1 -1 1 0 1 0
-1 -1 -1 1 1 0 0
Note that this is 18 possible patterns for any given row, so your matrix A can have 18^4 = 104,976 possible row patterns (quite a bit). You can generate every possible 4-wise row pattern index by using the functions NDGRID, CAT, and RESHAPE:
[indexSets{1:4}] = ndgrid(1:18);
indexSets = reshape(cat(5,indexSets{:}),[],4);
And indexSets will be a 104,976-by-4 matrix with each row containing one combination of 4 values between 1 and 18, inclusive, to be used as indices into rowPatterns to generate a unique matrix A. Now you can loop over each set of 4-wise row pattern indices and generate the matrix A using the functions TRIL, TRIU, EYE, and ZEROS:
for iPattern = 1:104976
A = rowPatterns(indexSets(iPattern,:),:); %# Get the selected row patterns
A = [tril(A,-1) zeros(4,1)] + ... %# Separate the 7-by-4 matrix into
[zeros(4,1) triu(A)] + ... %# lower and upper parts so you
[eye(4) zeros(4)]; %# can insert the diagonal ones
%# Store A in a variable or perform some computation with it here
end
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