Jökull - 01.12.1973, Page 43

which inserted in (10) gives
AA' a = b (21)
Since the matrix AA' is invertible we have a
unique solution
a = (AA')-i b (22)
which on the basis of (20) gives the solution
of our minimum problem and hence the gen-
eralized solution of the underdetermined pro-
blem defined by equations (10), viz.
x0 = A'(AA')-i b (23)
The composite matrix on the right of (23) is a
generalized inverse of A. Note that since A' is
n x m, AA' is m x m and, hence A'(AA')-1 is
11 x m. This operator transforms the m-vector b
mto the n-vector x0. The generalized inverse sim-
plifies to an ordinary inverse when m = n. To
verify that x0 is orthogonal to the null-space
N of A, we form the scalar product of x0 with
ar>y solution vector s of (13) and apply the
bilinear identity (11)
s - xo = s • A'(AA')-1 b = As • (AA')-ib = 0 (24)
The overdetermined, case
When m > n there are too many equations
to define a solution of (10). The hyperplanes
defined by each of the equations have no com-
mon point of intersection. As in the above case
°f m = 3^ n = g, we will again define a gener-
alized solution on the basis of the vector x0
whose end point S in n-space has the least
distance square sum from the m hyperplanes.
Since equations (10) have been normalized, x0
ls obtained as the solution of the following
niinimum problem
M = |Ax — b|2 = min. (25)
Using standard techniques again we replace x
ln (25) by the varied vector x c gx and form
= 2A8X ■ Ax - 2A8x • b = 0 (26)
c = 0
which with help of the adjoint A' reduces to
(A'Ax - A'b) ■ 8x = 0 (27)
Since 8x is an arbitrary vector the solution
vector x0 is obtained from
A'Ax0 = A'b (28)
and hence
x0 = (A'A)-1 A'b (29)
which is the generalized solution of the over-
determined problem. Note that A'A is n x n
and (A'A)-1 A' is n x m. Again the operator on
the right of (29) transforms the m-vector b into
the n-vector x0. It is a generalized inverse of
A which simplifies to the ordinary inverse when
m = n.
It is frequently of interest to introduce a
bias into the above procedure. We may want
to place unequal weights on the m equations
in (10). The numerical bias can be expressed
with the help of a diagonal m x m weight
matrix W with non-zero diagonal elements and
a trace (diagonal sum) equal to unity. Intro-
ducing W in equation (25) we now derive the
solution of
I V W (Ax —b)|2 =
____ (30)
| VwAx- Vwbj2 = min.
which on the basis of (29) has the solution
(31)
[(V W A)' Vw A]-1 ( V W A)' V W b
Because of the simplicity of the diagonal matrix
this expression reduces to
x0 = (A'WA)-1 A'Wb (32)
Equation (28) indicates that the generalized
solution x0 is obtained by operating on both
sides of (10) with the adjoint matrix A'. Since
the range of A' is orthogonal to the null-space
of A, this operation negates all components of
b which are in the null-space of A and thereby
opens the way for inverting A. It is evident
that A' is not the only matrix available for this
JÖKULL 23. ÁR 41