Styjun

I wonder if there is a way raise power of a matrix A as an array?

Assume that we have this matrix

A =

 5 4
 3 6

Then we repeat its shape.

>> repmat(A, 5, 1)
ans =

 5 4
 3 6
 5 4
 3 6
 5 4
 3 6
 5 4
 3 6
 5 4
 3 6

Now I want to rase the power so the long repeated matrix looks like this:

>> [A^1; A^2; A^3; A^4; A^5]
ans =

 5 4
 3 6
 37 44
 33 48
 317 412
 309 420
 2821 3740
 2805 3756
 25325 33724
 25293 33756

Is it possible to do that without a for loop in MATLAB/Octave?

Another option using arrayfun

B = cell2mat(arrayfun(@(x)A^x,1:5,'UniformOutput',0).')

Result:

B =
 5 4
 3 6
 37 44
 33 48
 317 412
 309 420
2821 3740
2805 3756
25325 33724
25293 33756

But a basic for-loop would probably be the fastest option in this case.

Benchmarking with octave:

tic
iif = @(varargin) varargin2*find([varargin1:2:end], 1, 'first')();
recPower = @(A, B, n, f) iif(n > 1, @() [B; f(A, A * B, n - 1, f)], true, @() B);
nPower = @(A, n) recPower(A, A, n, recPower);
for ii = 1:1000
% Calculate for arbitrary n.
 nPower(A, 5);
end
toc

Elapsed time is 1.023 seconds.

tic
for ii = 1:1000
 B = cell2mat(arrayfun(@(x)A^x,1:5,'UniformOutput',0).');
end
toc

Elapsed time is 4.8684 seconds.

tic
for ii = 1:1000
 B=[];
 for jj = 1:5
 B = [B;A^jj];
 end
end
toc

Elapsed time is 0.039371 seconds

EDIT

To also mention this in my answer: Recursion is not vectorization as of Matlab/Octave users usually might think of. I just had the idea of a recursive, anonymous function in my mind, and found the given task a nice small example to test the referenced solution on.

I was looking for recursive, anonymous functions and found this great answer. I incorporated the ideas from there to meet the expectations stated in the question, and came to this short code snippet.

% Input.
A = [5 4; 3 6]

% Set up recursive anonymous function.
iif = @(varargin) varargin2*find([varargin1:2:end], 1, 'first')();
recPower = @(A, B, n, f) iif(n > 1, @() [B; f(A, A * B, n - 1, f)], true, @() B);
nPower = @(A, n) recPower(A, A, n, recPower);

% Calculate for arbitrary n.
nPower(A, 5)

For explanations, please have a look at the linked answer.

Output:

A =

 5 4
 3 6

ans =

 5 4
 3 6
 37 44
 33 48
 317 412
 309 420
 2821 3740
 2805 3756
 25325 33724
 25293 33756

If you really want to use vectorization (which is IMO overkill in this case), you could also use the property:

A^n = P*D^n*P^-1 %A SHOULD BE a diagonalizable matrix

Where

[P,D] = eig(A) %the eigenvectors and eigenvalue

So

A = [5 4; 3 6]
n = 5;
% get the eigenvalue/eigenvector
[P,D]=eig(A);
% create the intermediate matrix
MD = diag(D).^[1:n];
MD = diag(MD(:));
% get the result
SP = kron(eye(n,n),P)*MD*kron(eye(n,n),P^-1);

With:

SP =

 5 4 0 0 0 0 0 0 0 0
 3 6 0 0 0 0 0 0 0 0
 0 0 37 44 0 0 0 0 0 0
 0 0 33 48 0 0 0 0 0 0
 0 0 0 0 317 412 0 0 0 0
 0 0 0 0 309 420 0 0 0 0
 0 0 0 0 0 0 2821 3740 0 0
 0 0 0 0 0 0 2805 3756 0 0
 0 0 0 0 0 0 0 0 25325 33724
 0 0 0 0 0 0 0 0 25293 33756

You still just need to extract those values. It could be interesting to use sparse matrix in this case to reduce memory usage.

Something like this:

SP = sparse(kron(eye(n,n),P))*sparse(MD)*sparse(kron(eye(n,n),P^-1));