MATH 372 -- Numerical Linear Algebra
QR-factorization example 
March 14, 2007

Consider the following 3 x 3 matrix


>> A = [2 1 1; 3 2 0; 4 1 -2]

A =

     2     1     1
     3     2     0
     4     1    -2

We first perform the QR factorization algorithm
"by hand".  The first step is to find a reflector
that takes the first column of A to a vector with
zeroes below the first entry.  We use the Corollary
we proved in class that for any two unequal vectors 
x, y with the same 2-norm there is a reflector taking the
first to the second.  Here x will the be first column
of A and y will be the vector [5.3852; 0; 0].
The reflector is determined by the unit vector u
in the direction of v = x - y:

>> norm([2; 3; 4],2)

ans =

    5.3852

>> v = [2; 3; 4] - [ 5.3852; 0; 0]

v =

   -3.3852
    3.0000
    4.0000

>> u = (1/norm(v,2))*v

u =

   -0.5606
    0.4968
    0.6625

The reflector is the following matrix:

>> Q1 = eye(3) - 2*u*u'

Q1 =

    0.3714    0.5571    0.7428
    0.5571    0.5063   -0.6583
    0.7428   -0.6583    0.1223

Multiplying Q1 times A we find the following, 
as expected:

>> Ahat = Q1*A

Ahat =

    5.3852    2.2283   -1.1142
    0.0000    0.9114    1.8736
    0.0000   -0.4514    0.4982

Now we continue with the same process on the 2 x 2
submatrix on the lower right:

>> norm([.9114; -.4514],2)

ans =

    1.0171

>> v = [.9114; -.4514] - [1.0171; 0]

v =

   -0.1057
   -0.4514

>> u = (1/norm(v,2))*v

u =

   -0.2280
   -0.9737

>> u2 = [0; -.2280; -.9737]

u2 =

         0
   -0.2280
   -0.9737

>> Q2 = eye(3) - 2*u2*u2'

The reflector Q2 will produce the final
zero below the diagonal:

Q2 =

    1.0000         0         0
         0    0.8960   -0.4440
         0   -0.4440   -0.8962

>> Q2*Ahat

ans =

    5.3852    2.2283   -1.1142
    0.0000    1.0171    1.4576
   -0.0001   -0.0001   -1.2783

(Note that we don't get exact zeroes because of round-off
errors(!))

>> R = Q2*Ahat

R =

    5.3852    2.2283   -1.1142
    0.0000    1.0171    1.4576
   -0.0001   -0.0001   -1.2783

The final factorization is given by:
>> Q = Q1'*Q2'

Q =

    0.3714    0.1694   -0.9130
    0.5571    0.7459    0.3651
    0.7428   -0.6441    0.1827

>> Q*R

ans =

    2.0000    0.9999    1.0002
    3.0000    2.0000   -0.0001
    4.0000    1.0000   -2.0000

(close!)   MATLAB also has a better built-in
QR-factorization routine.  It does the same
computation as above, except it makes the
diagonal entries in R negative rather than 
positive as above:


>> [QM,RM]=qr(A)

QM =

   -0.3714   -0.1695   -0.9129
   -0.5571   -0.7459    0.3651
   -0.7428    0.6442    0.1826


RM =

   -5.3852   -2.2283    1.1142
         0   -1.0171   -1.4578
         0         0   -1.2780