ON THE SCALAR FUNCTIONS OF HYPER COMPLEX

NUMBERS.

SECOND PAPER.

By HENRY TABER.
§ 1.

In this paper I shall denote by Yik, for i, j, k = 1, 2,...m, the constants of multiplication of a given non-nilpotent hyper complex number system (e1, eg,..em). We then have

In These Proceedings, vol. 41 (1905), p. 59, I have shown that there are two functions of the coefficients of any number

of the system (e1, e2,... em) constituting generalizations of the scalar function of quaternions, to which they reduce, becoming identical when m = 4, and, at the same time, the system (e1, e2, eз, es) is equivalent to the system constituted by the four units of quaternions. These functions, in designation the first and second scalar of A, are defined as follows:

and conform to theorem I given below. In this paper I shall employ these functions to establish a simple criterion for the existence of an

1 A number A

=

=

m

i=1

are of any hyper complex system (e1, e2, em) is idempotent if A2 A0; A is nil potent, if A 0 but AP = 0 for some positive integer p > 1. A system is nilpotent, if it contains no idempotent number; otherwise, non-nil potent. Every number of a nilpotent system is nilpotent. See B. Peirce, Am. Journ. Maths., 4, 113, (1881); cf. H. E. Hawkes, Trans. Am. Math. Soc., 3, 321 (1902).

invariant nilpotent sub system of (e1, e2,...em), and a method of determining the maximum invariant nilpotent sub system, if any exist. These results are embodied in theorem II.

=

Theorem I. Let Yijk, for i, j, k 1, 2,...m, be the constants of multiplication of any given hyper complex number system (e1, e2,

em).

Then both SA and S2 A are invariant to any linear transformation of the system: that is, if

+ Timem

(i = 1, 2, ... m), the determinant of transformation not being zero, and if

(e1, ez,

2 A sub system B1, B2, ... Bp of any hyper complex number system em) is said to be invariant if the product in either order of each number of (e1, e2, ...em) and each number of (B1, B2, ... Bp) belongs to the sub system, for which the necessary and sufficient conditions are

e¡ Bj = g'ri, B1 + 9'′2i, B2 + . . . + 9′mij Bp,

An invariant sub system (B1, B2, ... Bp) is an invariant nilpotent sub system if its units by themselves constitute a nilpotent system; and in that case is a maximum invariant nilpotent sub system if it contains every invariant nilpotent sub system of (e1, ez, em).

for every positive integer p; and conversely, if either

If A is idempotent, there are m S1A> 0 linearly independent numbers of the system satisfying the equation

AX = X,

in terms of which every number of the system satisfying this equation can be expressed linearly, also m S2 A > 0 linearly independent numbers satisfying the equation

XA = X,

in terms of which every solution of this equation can be expressed linearly.3

3 See paper by the author cited above, pp. 61, 69, and 70, also Trans. Am. Math. Soc., 5, 522, (1904).

and let the number system (e1, e2...em) contain at least one number satisfying the system of equations

(6)

S1 X ei = x1811; + x2 $1 €2¤; + + xm S1 em Ci = 0
(i = 1, 2, ... m).

...

we then have A1

= 0.

=

Let X B be any solution of equations (6). Then, by theorem I, B is nilpotent. Moreover, for any number A of (e1, ez,...em), both BA and AB are also solutions of equations (6). For, for any number

S1BY y1S1 Be1 + y2 S1 Bе2 + ... + Ym S1 Bem = 0;

in particular,

=

Since both BA and 4 B are solutions of equations (6), they are both

nilpotent.

Further, since, for 1≤i≤m, Be, is nilpotent, it follows from theorem I that S2 Be; 0, and thus any solution B of the system of equations (6) is also a solution of the system of equations

ei

=

=

(8) S2 Xe x1 S2 € 1 € ; + x2 S2 € 2 €; + ... + xm Szem li = 0
ei
(i = 1, 2, ... m),

By theorem I every solution of equations (8) is nilpotent. Let B' be any solution of this system of equations. Precisely as above, we may show that B' is nilpotent, and that both B'A and AB' are also solutions of these equations for any number A of the system (e1, e2, ... em); and, therefore, both B'A and A B' are nilpotent. Since, in particular, for 1 ≤i≤m, B'e, is nilpotent, it follows from theorem I that B' is a solution of the system of equations (6).

4

Let now the nullity of the determinant A1 be m', where 0<m'<m. There is then a set of just m' linearly independent numbers, B1, B2 ... Bm of the system (e1, e2...em) satisfying equations (6); therefore, just m' linearly independent numbers satisfying equations (8), whence it follows that the nullity of Ag is m'. For 1≤ j ≤ m', the product of B; in either order with any number A of the system is a solution of equations (6) and, therefore, both B, A and AB; are expressible linearly in terms of B1, B2 ... Bm'; otherwise, there is a set of more than m' linearly independent solutions of equations (6) which is contrary to supposition. Moreover, since

S1 (p1 B1 + p2 B2 + ... + Pm Bm) ei
B1+ B2+

= Pi Si Bies + p2 S1 B2е; +

(i

...

=

+Pm S1 Bmi = 0
1, 2, m),

...

every number linear in the B's is a solution of equations (6), and is, therefore, nilpotent. Whence it follows that B1, B2...Bm constitute an invariant nilpotent sub system of (e1, e2...em).

Further, the sub system (B1, B2 ... Bm) contains every invariant nilpotent sub system of (e1, e2... em), and is therefore the maximum invariant nilpotent sub system of the latter. For, let (C1, C2 . . . Cp) be any invariant nilpotent sub system of (e1, c2 ... em). Since every number of this sub system is nilpotent, in particular,

4 The nullity of a matrix or determinant of order m is m' if every (m' — 1)th minor (minor of order m — m' + 1) is zero but not every m'th minor (minor of order mm1). Nullity of order m' is equivalent to rank (Rang) m — m'.