ey (t) z 2,144 (+) + ay odo (5) du. ia icio core ey dp C6) T Tr, (+) (1)
where T, (+) is a residual vector.
A set of such linear combinations can in matrix notation be
written as
6 = 331,337 DA
= à,0,+ R5Q5 + ++... + aq. +R (2)
The matrix C is known, while the vectors a. and a; are to
be determined. In cases when the rank of the matrix C is
less than or equal to the number of base functions, the
vectors a; and q, may be chosen in such a way that the
residual matrix R is equal to zero.
There is an infinite number of solutions to this equation
system. The solution chosen in the K-L expansion is the one
where the base functions d; are normalized and orthogonal
and where the vectors a, are orthogonal. This can be
expressed as
3; d; = n-1 (3a)
di d; 0 , when i * j (3b)
8 a, =. 0 ,~when i #_3J (3c)
It is shown (Rao, 1965), that this solution has some very
favourable characteristics.
The residual matrix R is expressed as
i stat 5ra-7 = =T
Rz Cu 244) = 8505 = +... - & (4)
The size of the elements of a matrix can be expressed by its
Euclidian norm. The norm IRI is then estimated by
IRI* te (e;
i j
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