MATLAB: Toeplitz Matrix Generation from 2 MATRICES

MATLABteoplitz

Hello Everyone,
I know we can define a row and column vector then use the toeplitz function to generate a toeplitz matrix, but how can I do that when I have two matrices instead of vectors?
Suppose I have matrices A and B, and I want to generate a toeplitz matrix such that;
T = [B              0            0    ...  0;
     A*B            B            0    ...  0;
     A^2*B          A*B          B    ...  0;
     .              .            .    ...  .
     .              .            .    ...  .
     A^(n-1)*B      A^(n-2)*B    .    ...  B];
size(A) is KxK
size(B) is KxM

Best Answer

>> A = rand(3,3);
>> B = rand(3,5);
>> N = 4;
>> F = @(n)(A^n)*B;
>> C = arrayfun(F,0:N,'uni',0);
>> C = [{zeros(size(B))},C];
>> X = 1+tril(1+toeplitz(0:N));
>> M = cell2mat(C(X))
M =
   0.69956   0.86686   0.20666   0.46757   0.94893   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   0.46328   0.36229   0.56426   0.85822   0.71207   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   0.77021   0.83655   0.31248   0.91479   0.48724   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.04535   1.08015   0.66586   1.30393   1.19787   0.69956   0.86686   0.20666   0.46757   0.94893   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.19151   1.33153   0.52492   1.24584   1.20115   0.46328   0.36229   0.56426   0.85822   0.71207   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.13782   1.29286   0.44387   1.14332   1.08196   0.77021   0.83655   0.31248   0.91479   0.48724   0.00000   0.00000  0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.92239   2.11932   0.91061   2.08183   1.97011   1.04535   1.08015   0.66586   1.30393   1.19787   0.69956   0.86686  0.20666   0.46757   0.94893   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.89463   2.07432   0.92888   2.08761   1.95665   1.19151   1.33153   0.52492   1.24584   1.20115   0.46328   0.36229  0.56426   0.85822   0.71207   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   1.73168   1.89633   0.84706   1.90766   1.78390   1.13782   1.29286   0.44387   1.14332   1.08196   0.77021   0.83655  0.31248   0.91479   0.48724   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000  0.00000
   3.13506   3.43921   1.52166   3.43777   3.22857   1.92239   2.11932   0.91061   2.08183   1.97011   1.04535   1.08015  0.66586   1.30393   1.19787   0.69956   0.86686   0.20666   0.46757   0.94893   0.00000   0.00000   0.00000   0.00000  0.00000
   3.12822   3.43513   1.51057   3.42191   3.21683   1.89463   2.07432   0.92888   2.08761   1.95665   1.19151   1.33153  0.52492   1.24584   1.20115   0.46328   0.36229   0.56426   0.85822   0.71207   0.00000   0.00000   0.00000   0.00000  0.00000
   2.85612   3.13678   1.37809   3.12326   2.93605   1.73168   1.89633   0.84706   1.90766   1.78390   1.13782   1.29286  0.44387   1.14332   1.08196   0.77021   0.83655   0.31248   0.91479   0.48724   0.00000   0.00000   0.00000   0.00000  0.00000
   5.15716   5.66180   2.49328   5.64461   5.30487   3.13506   3.43921   1.52166   3.43777   3.22857   1.92239   2.11932  0.91061   2.08183   1.97011   1.04535   1.08015   0.66586   1.30393   1.19787   0.69956   0.86686   0.20666   0.46757  0.94893
   5.13621   5.63814   2.48462   5.62330   5.28412   3.12822   3.43513   1.51057   3.42191   3.21683   1.89463   2.07432  0.92888   2.08761   1.95665   1.19151   1.33153   0.52492   1.24584   1.20115   0.46328   0.36229   0.56426   0.85822  0.71207
   4.68804   5.14614   2.26789   5.13272   4.82305   2.85612   3.13678   1.37809   3.12326   2.93605   1.73168   1.89633  0.84706   1.90766   1.78390   1.13782   1.29286   0.44387   1.14332   1.08196   0.77021   0.83655   0.31248   0.91479  0.48724
And checking:
>> B % equal to main diagonal (e.g. top left corner and bottom right corner):
B =
   0.69956   0.86686   0.20666   0.46757   0.94893
   0.46328   0.36229   0.56426   0.85822   0.71207
   0.77021   0.83655   0.31248   0.91479   0.48724
>> A^N*B % equal to bottom left corner
ans =
   5.1572   5.6618   2.4933   5.6446   5.3049
   5.1362   5.6381   2.4846   5.6233   5.2841
   4.6880   5.1461   2.2679   5.1327   4.8230
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