Page 77 - Compact Numerical Methods For Computers

Chapter 5

SOME COMMENTS ON THE FORMATION OF THE
T
CROSS-PRODUCTS MATRIX A A

Commonly in statistical computations the diagonal elements of the matrix
T
(A A) -1 (5.1)
are required, since they are central to the calculation of variances for parameters
T
estimated by least-squares regression. The cross-products matrix A A from the
singular-value decomposition (2.53) is given by
T T T 2 T
A A = VSU USV = VS V . (5.2)
T
This is a singular-value decomposition of A A, so that
T + + 2 T
(A A) = V(S ) V . (5.3)
If the cross-products matrix is of full rank, the generalised inverse is identical to
the inverse (5.1) and, further,
+ -1
S = S . (5.4)
Thus we have
T -1 - 2 T
(A A) = VS V . (5.5)
The diagonal elements of this inverse are therefore computed as simple row
norms of the matrix
- l
VS . (5.6)
In the above manner the singular-value decomposition can be used to compute
the required elements of the inverse of the cross-products matrix. This means that
the explicit computation of the cross-products matrix is unnecessary.
T
Indeed there are two basic problems with computation of A A. One is induced
T
by sloppy programming practice, the other is inherent in the formation of A A.
The former of these occurs in any problem where one of the columns of A is
constant and the mean of each column is not subtracted from its elements. For
instance, let one of the columns of A (let it be the last) have all its elements equal
to 1. The normal equations (2.22) then yield a cross-products matrix with last row
(and column), say the nth,
(5.7)
But

(5.8)

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