304

THE FACTORS OF THE MIND

proper conditions are observed, the procedure will of itself ensure
that the resultant transformation is orthogonal. The above table
lends itself to a simple illustration showing how the peculiarities of
the transformation matrix arise.

(i) We begin with the bottom row of -F&, since this contains the
largest number of negative saturations. These can all be reduced
to zero by subtracting suitable submultiples of the first column
(which contains positive figures throughout) from all the other
columns in turn. This means post-multiplying Fb by a matrix of
the type :

' t _& _& ..."

o
o

/«!

I

0

o
I

I
V2
A/6~

Vis
V^S
V15

0
i
0
o

0
o
I
0

0
o
o
I

Since/42,/43 • • - are negative, this produces a row of positive figures
in the top line.

(ii) But this first operation inevitably raises the zero saturations
in the top right-hand triangle of the initial matrix to positive values.
If we are to maximize the zeros, these must be abolished. We
proceed as before ; and now the operation will be equivalent to
multiplying by a matrix including a corresponding triangle of
negative multipliers, viz. :

I O
0 I

0 0

o
I

I —

1/6

i

(iii) When all the negative saturations and all the eliminable
positive saturations have been annulled in this way, we multiply by
a diagonal matrix:

1 2

o

3
V3

o

o
o

o

I

o _.__

o

2