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########################################################################
##
## Copyright (C) 1995-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## This file is part of Octave.
##
## Octave is free software: you can redistribute it and/or modify it
## under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## Octave is distributed in the hope that it will be useful, but
## WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with Octave; see the file COPYING. If not, see
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{k} =} commutation_matrix (@var{m}, @var{n})
## Return the commutation matrix
## @tex
## $K_{m,n}$
## @end tex
## @ifnottex
## K(m,n)
## @end ifnottex
## which is the unique
## @tex
## $m n \times m n$
## @end tex
## @ifnottex
## @var{m}*@var{n} by @var{m}*@var{n}
## @end ifnottex
## matrix such that
## @tex
## $K_{m,n} \cdot {\rm vec} (A) = {\rm vec} (A^T)$
## @end tex
## @ifnottex
## @math{K(m,n) * vec(A) = vec(A')}
## @end ifnottex
## for all
## @tex
## $m\times n$
## @end tex
## @ifnottex
## @math{m} by @math{n}
## @end ifnottex
## matrices
## @tex
## $A$.
## @end tex
## @ifnottex
## @math{A}.
## @end ifnottex
##
## If only one argument @var{m} is given,
## @tex
## $K_{m,m}$
## @end tex
## @ifnottex
## @math{K(m,m)}
## @end ifnottex
## is returned.
##
## See @nospell{Magnus and Neudecker} (1988), @cite{Matrix Differential
## Calculus with Applications in Statistics and Econometrics}.
## @end deftypefn
function k = commutation_matrix (m, n)
if (nargin < 1)
print_usage ();
else
if (! (isscalar (m) && m == fix (m) && m > 0))
error ("commutation_matrix: M must be a positive integer");
endif
if (nargin == 1)
n = m;
elseif (! (isscalar (n) && n == fix (n) && n > 0))
error ("commutation_matrix: N must be a positive integer");
endif
endif
## It is clearly possible to make this a LOT faster!
k = zeros (m * n, m * n);
for i = 1 : m
for j = 1 : n
k((i - 1) * n + j, (j - 1) * m + i) = 1;
endfor
endfor
endfunction
%!test
%! c = commutation_matrix (1,1);
%! assert (c,1);
%!test
%! A = rand (3,5);
%! vc = vec (A);
%! vr = vec (A');
%! c = commutation_matrix (3,5);
%! assert (c*vc, vr);
%!test
%! A = rand (4,6);
%! vc = vec (A);
%! vr = vec (A');
%! c = commutation_matrix (4,6);
%! assert (c*vc, vr);
%!error <M must be a positive integer> commutation_matrix (0,0)
%!error <N must be a positive integer> commutation_matrix (1,0)
%!error <M must be a positive integer> commutation_matrix (0,1)
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