From: Dave Rusin <rusin@math.niu.edu>
Subject: Re: determinant of non square matrix
Date: Tue, 09 May 2000 10:39:39 +0200
To: Didier Bizzarri <D.Bizzarri@ulg.ac.be>

>I would like to know whether there is an existing theory
>about the determinant of non square matrice?

The determinant is not defined for non-square matrices.

There are a number of useful properties of determinants, some of which
may be generalized for non-square matrices as well. But you need to decide
which features of the determinant are of interest. For example, since
determinants are multiplicative, and since orthogonal matrices have
determinant +-1 , the determinant is the product of the singular values
of the matrix (that is, we may write  A = U D V  with  U, V   orthogonal
and  D  diagonal; then  det(A) = prod( D_i ). )  Well, non-square matrices
have a singular value decomposition too, with  D  a non-square diagonal
matrix. A perfectly reasonable generalization of the determinant is
then  'det(A)' = prod( D_i )  for all matrices, square or otherwise.
Whether or not this is the "right" definition depends on what you hoped
to accomplish with your generalization.

[deletia --djr]