Jökull - 01.12.1973, Page 42

represent a set of m linear equations with n
unknowns. The coefficients aik are assumed to
be real.
In the following it is convenient to assume
that the equations have been normalized such
that the h-norm of the row-vectors is unity,
viz.
If jrj| zjí= 1 this can always be achieved by
dividing each equation by jr±J. The underdeter-
mined, well posed and overdetermined cases are
then characterized by m < n, m = n, and m > n
respectively. It is also convenient to rewrite
(8) in vector-matrix language. Let x be the un-
known n-vector (xk), b the given m-vector (bj),
and A = (alk) be the m x n matrix of the en-
tries aik where i = 1, 2,. . ., m, and k=l,2,
. . ., n. The system (8) can then be written
Ax = b (10)
Let A' be the n x m transpose or adjoint of
A and x • y denote the scalar product of two
vectors x and y. The adjoint satisfies the
identity
x • Ay = A'x • y (11)
for any m-vector x and n-vector y. The product
matrix AA' is m x m and A'A is n x n. These
two matrices are symmetric. In the following we
will assume that our basic matrix A is such that
both AA' and A'A are non-singular and hence
invertible. The identity (11) shows that the
range of A' is orthogonal to the null-space of
A and vice versa. Hence, any solution of the
homogeneous equations (10) has to be ortho-
gonal to the range of the adjoint A'.
The underdetermined case
Let equations (10) represent an underdeter-
mined case where m < n. Adopting the solu-
tion method indicated by equations (2) and (3)
above, we will assume that the solution n-
vectors of (10) can be represented
x = x0 + s (12)
40 JÖKULL 23. ÁR
where x0 is the least 12-norm vector satisfying
(10) and the vector s lies wholly within the
null-space N of A, viz.
As = 0 (13)
for any s in N. The decomposition (12) prc-
supposes the orthogonality of x0 and s, which
will be verified below, and hence,
|X|2=|X0|2+ |S|2 (14)
The vector x0 is consequently to be found as
the solution of the following minimum pro-
blem
[x|2 = min. (15)
for all x which satisfy equations (10). This
problem is solved by a standard variational
technique, that is, we minimize the following
expression
M = |x|2 + 2a • (b — Ax) (16)
where a is the Lagrange m-vector multiplier
which is to be determined. The factor 2 in
equation (16) is introduced for convenience.
Let <3x be an arbitrary small n-vector and c a
scalar. The vector gx is introduced into (16) as
the variation of x, that is, we replace x in (16)
by x + c ðx and minimize M with respect to
c at c = 0, viz.
= 2x-8x-2a-A8x = 0 (17)
c = 0
The second term on the right of (17) can be
written with the help of the adjoint A'
a-A8x = A'a-8x (18)
and hence (17) can be rewritten
(x — A' a) • 8x = 0 (19)
This equation has to hold for an arbitrary 8x,
viz. our solution
x0 = A' a
3M
9c
(20)