| | DEFUN_DLD (schur, args, nargout,"-*- texinfo -*-\n\
@deftypefn {Loadable Function} {@var{s} =} schur (@var{a})\n\
@deftypefnx {Loadable Function} {[@var{u}, @var{s}] =} schur (@var{a}, @var{opt})\n\
@cindex Schur decomposition\n\
The Schur decomposition is used to compute eigenvalues of a\n\
square matrix, and has applications in the solution of algebraic\n\
Riccati equations in control (see @code{are} and @code{dare}).\n\
@code{schur} always returns\n\
@iftex\n\
@tex\n\
$S = U^T A U$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{s = u' * a * u}\n\
@end ifnottex\n\
where\n\
@iftex\n\
@tex\n\
$U$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{u}\n\
@end ifnottex\n\
is a unitary matrix\n\
@iftex\n\
@tex\n\
($U^T U$ is identity)\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
(@code{u'* u} is identity)\n\
@end ifnottex\n\
and\n\
@iftex\n\
@tex\n\
$S$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{s}\n\
@end ifnottex\n\
is upper triangular. The eigenvalues of\n\
@iftex\n\
@tex\n\
$A$ (and $S$)\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{a} (and @code{s})\n\
@end ifnottex\n\
are the diagonal elements of\n\
@iftex\n\
@tex\n\
$S$.\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{s}.\n\
@end ifnottex\n\
If the matrix\n\
@iftex\n\
@tex\n\
$A$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{a}\n\
@end ifnottex\n\
is real, then the real Schur decomposition is computed, in which the\n\
matrix\n\
@iftex\n\
@tex\n\
$U$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{u}\n\
@end ifnottex\n\
is orthogonal and\n\
@iftex\n\
@tex\n\
$S$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{s}\n\
@end ifnottex\n\
is block upper triangular\n\
with blocks of size at most\n\
@iftex\n\
@tex\n\
$2\\times 2$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{2 x 2}\n\
@end ifnottex\n\
along the diagonal. The diagonal elements of\n\
@iftex\n\
@tex\n\
$S$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{s}\n\
@end ifnottex\n\
(or the eigenvalues of the\n\
@iftex\n\
@tex\n\
$2\\times 2$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{2 x 2}\n\
@end ifnottex\n\
blocks, when\n\
appropriate) are the eigenvalues of\n\
@iftex\n\
@tex\n\
$A$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{a}\n\
@end ifnottex\n\
and\n\
@iftex\n\
@tex\n\
$S$.\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{s}.\n\
@end ifnottex\n\
\n\
The eigenvalues are optionally ordered along the diagonal according to\n\
the value of @code{opt}. @code{opt = \"a\"} indicates that all\n\
eigenvalues with negative real parts should be moved to the leading\n\
block of\n\
@iftex\n\
@tex\n\
$S$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{s}\n\
@end ifnottex\n\
(used in @code{are}), @code{opt = \"d\"} indicates that all eigenvalues\n\
with magnitude less than one should be moved to the leading block of\n\
@iftex\n\
@tex\n\
$S$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{s}\n\
@end ifnottex\n\
(used in @code{dare}), and @code{opt = \"u\"}, the default, indicates that\n\
no ordering of eigenvalues should occur. The leading\n\
@iftex\n\
@tex\n\
$k$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{k}\n\
@end ifnottex\n\
columns of\n\
@iftex\n\
@tex\n\
$U$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{u}\n\
@end ifnottex\n\
always span the\n\
@iftex\n\
@tex\n\
$A$-invariant\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{a}-invariant\n\
@end ifnottex\n\
subspace corresponding to the\n\
@iftex\n\
@tex\n\
$k$\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{k}\n\
@end ifnottex\n\
leading eigenvalues of\n\
@iftex\n\
@tex\n\
$S$.\n\
@end tex\n\
@end iftex\n\
@ifnottex\n\
@code{s}.\n\
@end ifnottex\n\
@end deftypefn") |
