Page 36 - Compact Numerical Methods For Computers

26 Compact numerical methods for computers
+
is the annihilator of A b, thus ensuring that the two contributions (that is, from b
T +
and g) to x x are orthogonal. This requirement imposes on A the further
conditions
+ + +
A AA = A (2.44)
+ T +
(A A) = A A. (2.45)
The four conditions (2.40), (2.41), (2.44) and (2.45) were proposed by Penrose
(1955). The conditions are not, however, the route by which A + is computed.
Here attention has been focused on one generalised inverse, called the Moore-
Penrose inverse. It is possible to relax some of the four conditions and arrive at
other types of generalised inverse. However, these will require other conditions to
be applied if they are to be specified uniquely. For instance, it is possible to
consider any matrix which satisfies (2.40) and (2.41) as a generalised inverse of A
since it provides, via (2.33), a least-squares solution to equation (2.14). However,
in the rank-deficient case, (2.36) allows arbitrary components from the null space
of A to be added to this least-squares solution, so that the two-condition general-
ised inverse is specified incompletely.
Over the years a number of methods have been suggested to calculate ‘generalised
inverses’. Having encountered some examples of dubious design, coding or appli-
cations of such methods, I strongly recommend testing computed generalised inverse
matrices to ascertain the extent to which conditions (2.40), (2.41), (2.44) and (2.45)
are satisfied (Nash and Wang 1986).

2.5. DECOMPOSITIONS OF A MATRIX
In order to carry out computations with matrices, it is common to decompose
them in some way to simplify and speed up the calculations. For a real m by n
matrix A, the QR decomposition is particularly useful. This equates the matrix A
with the product of an orthogonal matrix Q and a right- or upper-triangular
matrix R, that is
A = QR (2.46)
where Q is m by m and
T
Q Q = QQ T = 1 m (2.47)
and R is m by n with all elements
R = 0 for i > j. (2.48)
ij
The QR decomposition leads to the singular-value decomposition of the matrix A
if the matrix R is identified with the product of a diagonal matrix S and an ortho-
T
gonal matrix V , that is T
R = SV (2.49)
where the m by n matrix S is such that

S = 0 for i j (2.50)
ij
and V, n by n, is such that
T T
V V = VV = 1 . (2.5 1)
n