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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{d} =} duplication_matrix (@var{n})
## Return the duplication matrix
## @tex
## $D_n$
## @end tex
## @ifnottex
## @nospell{@math{Dn}}
## @end ifnottex
## which is the unique
## @tex
## $n^2 \times n(n+1)/2$
## @end tex
## @ifnottex
## @math{N^2}-by-@math{N*(N+1)/2}
## @end ifnottex
## matrix such that
## @tex
## $D_n * {\rm vech} (A) = {\rm vec} (A)$
## @end tex
## @ifnottex
## @nospell{@code{Dn * vech (A) = vec (A)}}
## @end ifnottex
## for all symmetric
## @tex
## $n \times n$
## @end tex
## @ifnottex
## @math{N}-by-@math{N}
## @end ifnottex
## matrices
## @tex
## $A$.
## @end tex
## @ifnottex
## @math{A}.
## @end ifnottex
##
## See @nospell{Magnus and Neudecker} (1988), @cite{Matrix Differential
## Calculus with Applications in Statistics and Econometrics}.
## @end deftypefn
function d = duplication_matrix (n)
if (nargin < 1)
print_usage ();
endif
if (! (isscalar (n) && n > 0 && n == fix (n)))
error ("duplication_matrix: N must be a positive integer");
endif
d = zeros (n * n, n * (n + 1) / 2);
## It is clearly possible to make this a LOT faster!
count = 0;
for j = 1 : n
d((j - 1) * n + j, count + j) = 1;
for i = (j + 1) : n
d((j - 1) * n + i, count + i) = 1;
d((i - 1) * n + j, count + i) = 1;
endfor
count += n - j;
endfor
endfunction
%!test
%! N = 2;
%! A = rand (N);
%! B = A * A';
%! C = A + A';
%! D = duplication_matrix (N);
%! assert (D * vech (B), vec (B), 1e-6);
%! assert (D * vech (C), vec (C), 1e-6);
%!test
%! N = 3;
%! A = rand (N);
%! B = A * A';
%! C = A + A';
%! D = duplication_matrix (N);
%! assert (D * vech (B), vec (B), 1e-6);
%! assert (D * vech (C), vec (C), 1e-6);
%!test
%! N = 4;
%! A = rand (N);
%! B = A * A';
%! C = A + A';
%! D = duplication_matrix (N);
%! assert (D * vech (B), vec (B), 1e-6);
%! assert (D * vech (C), vec (C), 1e-6);
%!error <Invalid call> duplication_matrix ()
%!error duplication_matrix (0.5)
%!error duplication_matrix (-1)
%!error duplication_matrix (ones (1,4))
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