I have been doing Gilbert Strang’s linear algebra assignments, some of which require you to write short scripts in MatLab, though I use GNU Octave (which is kind of like a free MatLab). I was trying out this problem:
To solve this quickly, it would have been nice to have a function that would give a list of permutation matrices for every n-sized square matrix, but there was none in Octave, so I wrote a function permMatrices which creates a list of permutation matrices for a square matrix of size n.
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| % function to generate permutation matrices given the size of the desired permutation matrices | |
| function x = permMatrices(n) | |
| x = zeros(n,n,factorial(n)); | |
| permutations = perms(1:n); | |
| for i = 1:size(x,3) | |
| x(:,:,i) = eye(n)(permutations(i,:),:); | |
| end | |
| endfunction |
For example:
The MatLab / Octave code to solve this problem is shown below:
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| % Solution for part (a) | |
| p = permMatrices(3); | |
| n = size(p,3); % number of permutation matrices | |
| v = zeros(n,1); % vector of zeros with dimension equalling number of permutation matrices | |
| % check for permutation matrices other than identity matrix with 3rd power equalling identity matrix | |
| for i = 1:n | |
| if p(:,:,i)^3 == eye(3) | |
| v(i,1) = 1; | |
| end | |
| end | |
| v(1,1) = 0; % exclude identity matrix | |
| ans1 = p(:,:,v == 1) | |
| % Solution for part (b) | |
| P = permMatrices(4); | |
| m = size(P,3); % number of permutation matrices | |
| t = zeros(m,1); % vector of zeros with dimension equalling number of permutation matrices | |
| % check for permutation matrices with 4th power equalling identity matrix | |
| for i = 1:m | |
| if P(:,:,i)^4 == eye(4) | |
| t(i,1) = 1; | |
| end | |
| end | |
| % print the permutation matrices | |
| ans2 = P(:,:,t == 0) |
Output:
