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Consider
n
linear elements
e
i
, each of which is actively connected to all
others and to itself. We have a perfect connection matrix, all rows and
columns being non-zero. Let r(
i
) and s(
i
) represent response and external
stimulus of element
e
i
respectively and permit a certain fraction
a
ij
of the
response of element
e
i
to contribute to the stimulus of element
e
j
.The
response of element
e
1
is under these circumstances clearly
()
=
()
+
()
+
()
++
()
rs r
1
1
a
1
a
2
...
a
r
n
,
(64)
11
21
n
1
or in general for the
j
th element:
n
Â
()
=
()
+
()
rs
j
j
a
r
i
,
(65)
ij
i
=
1
where for simplicity s and r are expressed in the same arbitrary units and
where the coefficients
a
ij
form again a square matrix which we will call the
numerical interaction matrix
(66)
Aa
n
=
.
ij
n
In order to obtain a solution for the
n
unknowns r(
j
) in terms of all stimuli
s(1), s(2),...,s(
n
), we first express all stimuli s(
j
) in terms of the various
responses r(1), r(2),...,r(
n
). From (64) we have for s(1)
()
=-
(
) ( )
-
()
-
()
s
11
a
r
1
a
r
2
s r
3
...
11
21
31
or in general for the
j
th element
n
Â
()
=
()
s
j
s
r
i
,
(67)
ij
i
=
1
where the
s
ij
form again a square matrix
S
n
which we will call the stimulus
matrix
Ss
n
=
(68)
ij
n
with
-
a
for
for
i
π
j
,
.
Ó
ij
s
=
(69)
ij
(
)
1
-
a
i
=
j
ii
In the formalism of matrix algebra the
n
values for s as well as for r rep-
resent
n
-dimensional vectors (column matrices) and (67) can be formally
represented by:
s
=
S
r
.
(70)
n
n
n
In order to find r
n
expressed in terms of s
n
one “simply” inverts the
matrix
S
n
and obtains
r
=
S
-1
s
,
(71)
n
n
n
which implies solving the
n
equations in (67) for the
n
unknowns r(1), r(2),
...,r(
n
). We introduce the response matrix
R
n
, defined by
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