MATLAB: Vectorization of matrices power

MATLABmatricesmatrixmatrix manipulation

Hi all,
Let A a matrix n*n and let N an integer. I wish to create a matrix of power of the form AN = [I A A^2,…,A^N], where I=eye(n), without resorting to a "for" cycle. I have tried to take a look at several commands, including cumprod, kron, etc, and trying to combine them, but I failed. I made it only for the scalar case, i.e. n=1.
After that, I wish to create a matrix
AA = [I 0 0 0;A I 0 0;A^2 A I 0;A^3 A^2 A I]
without using any "for" cycle. I have noticed that the first column of AA is equal to AN' (if it may help). Any hints? Thanks.

Best Answer

>> [AN,T]=matpowers(diag([2,3]),3),
AN =
       1     0     2     0     4     0     8     0
       0     1     0     3     0     9     0    27
T =
       1     0     0     0     0     0     0     0
       0     1     0     0     0     0     0     0
       2     0     1     0     0     0     0     0
       0     3     0     1     0     0     0     0
       4     0     2     0     1     0     0     0
       0     9     0     3     0     1     0     0
       8     0     4     0     2     0     1     0
       0    27     0     9     0     3     0     1
function  [AN,T]=matpowers(A,c)
   persistent pcell N
   if isempty(pcell), pcell={eye(size(A))}; end
   if nargin>1, N=c; end
   if ~isempty(N) && N>0
       pcell=[pcell,{pcell{end}*A}];
       N=N-1;
      AN=matpowers(A); 
   else
       AN=pcell; pcell=[]; N=[];
   end
   if nargout>1
    z=[AN,{zeros(size(A))}];
    T=toeplitz(1:c+1,[1,ones(1,c)*(c+2)]);
    T=cell2mat(z(T)) ;
    AN=cell2mat(AN);
   end
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