Dear Tom
Finding the well-posed question is part of the answer.
I am asking because I would like to impose structure on an estimated transition matrix in a state space model. Both the estimation and the interpretation of the results may benefit from this. At least the transition matrix should have a real eigenvector with a real positive eigenvalue. It may even be constrained to be orthogonal.
Experience has shown that it may be convenient to impose such parameter constraints via a transformation of an unconstrained parameter space. Typically, constrained x > 0 is parametrized as exp(y) with y unconstrained.
Following some tracks in wikipedia I came across the paper, http://www.ams.org/notices/200705/fea-mezzadri-web.pdf, which partly gives the solution I need in terms of a mapping from unit spheres (S^0 x S^1 x S^N) into an orthonormal matrix defined by Householder reflections. A mapping from R^n into S^n is given as:
u' = 4/(x'x + 4)*(x' ~ - 2) + (0' ~ 1)
I guess this is not a standard piece of Ox?
Thanks for the hints.
Øyvind
>>> Tom Petersen <[log in to unmask]> 2/8/2010 3:41 pm >>>
Dear Öyvind,
I'm sorry, but your question is ill-posed.
You can _decompose_ such a matrix into, for example, one triangular and
one orthonormal using QR or LQ decomposition, or into two orthonormal
and one diagonal using SVD (singular value decomposition); but the
orthonormal ones do not replace the original triangular matrix by
themselves.
Read more on
http://math.u-bourgogne.fr/monge/bibliotheque/ebooks/csa/htmlbook/node36.html
or Wikipedia: http://en.wikipedia.org/wiki/Orthogonal_matrix,
http://en.wikipedia.org/wiki/Matrix_decomposition.
The question is, what is your purpose?
/Tom
Oyvind Hoveid wrote:
> Dear Ox programmers
>
> Do you know of a function which computes an orthonormal matrix ( A * A' = I) from a lower triangular matrix of parmeters?
>
> Thanks in advance
>
> Øyvind
>
>
>
>
>
>
> -------------------------------------------------------
> Oyvind Hoveid
>
> Norwegian Agricultural Economics Research Institute
> ++47 2236 7264
> [log in to unmask]
>
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