80 
MATHEMATICS: F. L. HITCHCOCK 
Proc. N. a. S. 
The choice of the term AjxBj to be freed from matrix coefficients affects 
the convergence of the solution. 
2. Transformation from Matrices to Double Dyadics. — By Gibbs' double 
multiplication we may set up explicit formulas for the coefficients Wi . . . m„ 
of the Hamilton- Cay ley equation (3). It is known that any matrix A 
is equivalent to a dyadic Saa' where a and a' are vectors. Any matrix 
term of the form AxB is equivalent to a sum of terms of the form aa'.X.bb' 
where X is the dyadic equivalent to x. We now introduce the double dot 
product defined by the rule ab:xy = a.xb.y, and may write 
aa'.X.bb' = ab'a'b:X. (4) 
Any term AxB is therefore equivalent to a sum of terms of the form 
MN:X where M and N are dyadics, and the linear matrix equation (2) 
is equivalent to 
<^:X = SMNiX = C (5) 
where <^ is a dyadic whose antecedents and consequents are dyadics, say 
a double dyadic. 
3. Dyadics Treated Like Vectors in Space of Higher Dimensions, — Adopting 
unit vectors Ci . . . e„ such that ^i.^k is unity when subscripts are equal, 
otherwise zero, any dyadic M may be expanded in terms of the dyads 
6,6^^; and if we agree on some definite order, no matter what, among these 
dyads we may set = 6,6^^, lettering r run from i to as i and k each run 
from I to N. With n =N^, any dyadic may be expanded 
= Emti + Eam^z + . . . + E„m/„ (6) 
where m^ etc. are scalars. The double dyadic (p may be expanded as 
<p = EiMi + E2M2 + . . . + E„M„. (7) 
Now e^e^feie^e^ is unity when both i — r and k = s, otherwise' zero. This 
is the same assaying E^rE^ is unity when subscripts are equal, otherwise 
zero. It is immediately evident that the double dyadic (p will behave with 
reference to double dot product as would an ordinary dyadic in n- dimen- 
sional space with reference to ordinary dot product, and equally evident 
that the whole of the usual matrix theory will apply to <p with no essential 
change. 
4. Formation of the Hamilton-Cayley Equation. — Since (p:X = d{x) the 
Hamilton-Cayley equation for <p will be the same as that for 0. It re- 
mains to show how to calculate the coefficients mi . . m„ of this equation. 
Let Ml, M2, . . . , and Ni, N2, . . . , be any dyadics : 
Definition. — ^The scalar ((MiNi, M2N2, . . . , M^N^))^ is defined to be 
the sum of terms computed as follows: the leading term is the product 
of double dot products Mi:NiM2:N2 M^:N^; the other terms 
are of like form and are obtained from the leading term by keeping the 
antecedents Mi ... . M^ fixed in position while the consequents Ni . . . 
are permuted in all ways. The sign of any term is plus or minus according as 
the number of simple interchanges needed to form that term is even or odd. 
