From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: QR Factorization
Date: 23 Nov 2001 15:11:25 GMT
Newsgroups: sci.math.num-analysis,sci.math
Summary: Computing SVD numerically (with Matlab)

In article <7da9efed.0111230552.65163d71@posting.google.com>,
 MajorSetback@excite.com (PeterOut) writes:
snip 
|> According to ``Matrix Analysis'' by Roger Horn and Charles Johnson,
|> the output of the factorization is an m-by-n matrix (Q) with
|> orthonormal columns and an n-by-n upper matrix (R).  However, the
|> output in MATLAB consists of an m-by-m unitary matrix (Q) and an
|> m-by-n matrix (R) with the first n rows forming an upper triangular
|> matrix.  I was wondering if anyone could explain the discrepancy.
|> 
|> Many thanks in advance,
|> Peter.

this simply depends on convention. you may write

    A = Q R       A is m times n , m>=n,  Q is m times n, R is n times n
the simplest way to obtain Q is Gram-Schmidt orthonormalization 
(numerically unstable) or modified Gram-Schmidt orthonormalization 
(compute R rowwise), much better but again not as fine as 
Householder transformations. But here you originally have 

    U_n U_{n-1}   U_1 A  =  [ R ]
                            [ O ]

where the U_i are Householder reflectors and the "O" is a (m-n) times n zero
matrix. Now, setting 
    Q' = U_n U_{n-1} ... U_1
you get 
         
       A = Q [R]
             [O]       

and this is what MATLAB gives you . You can recover (*) by deleting the "O"
part and the last m-n columns of Q.
Normally Q is _not_ returned by QR-codes, rather the reflection normals which
define the U_i (this information suffices anyway).

hope that helps
peter