How to write a circulant matrix , where the elements are matrices? Thanks!

Hi， I want to write a circulant matrix and the element in the matrix is a matrix again. Is there simple way to do it. For example, if I want to write block-diag matrix, it is easy, but how can extend this to the circulant case.

Thanks in advance!

Answer 1

[Wouter]

Suppose the inner matrices have size Nin and the outer matrix size Nout

    for n=1:Nout
for m=1:Nout
[qn,rn]=quorem(n,Nin);
[qm,rm]=quorem(m,Nin);
outermatrix(n,m)=...
end
end

qn/qm will give you the index of which inner matrix to use, and rn, rm which element within this inner matrix.

Answer 2

[Chris Turnes]

Supposing the matrices you want to use in a circulant pattern are stored as a M x N x K array (where M x N is the size of each individual matrix block and you want a K x K circulant pattern), then one way to do it is as follows:

    % Original matrix
    A = randn(M,N,K);
    % Coefficients for block circulant
    Ac = permute(A, [1 3 2]);
    Ac = cat(2, Ac(:, 2:K, :), Ac);
    % Create the block-circulant matrix using convn
    I = reshape(eye(K), [1 K 1 K]);
    C = convn(I, Ac);
    C = reshape(C(:, K:(2*K-1), :, :), [M*K, N*K]);

This isn't the most memory efficient way to do it, but it's probably the fastest unless you start playing games with fftn. The idea here is to first permute A so that the "block" dimension is 2nd out of 3. Then, you extend Ac so that it's M x 2K-1 x N in such a way that the center portion of the convolution will give you a block circulant matrix. Next, you create an identity matrix to convolve against -- but you only want it to have non-scalar size in the block dimension, so it is reshaped to a 1 x K x 1 x K. Then you call convn and select the central M x K x N x K block. Once that's done, you can reshape it as an MK x NK matrix and you have your block circulant.

You don't have to permute A -- you can instead keep everything the dimension it is and permute C at the end. But, you'll be permuting a larger array, so I think it's preferable to work on the coefficient array in the beginning.

[Chris Turnes]

Alternatively, if you don't feel like jumping through these hoops, just loop and use circshift:

    C = zeros(M, K, N, K);
    Ac = permute(A, [1 3 2]);
    for i = 1:K
        C(:,:,:,i) = circshift(Ac, (i-1), 2);
    end
    C = reshape(C, [M*K, N*K]);

Answer 3

[Matt J]

   N=5; %size of circulant matrix
   blocks={A,B,C,D,...};  %list of sub-matrices as cell array

   idx = toeplitz(1:N,[1,N:-1:2])  ;

   blockmatrix = cell2mat(  blocks(idx) );