The Scalar Functions of Hypercomplex Numbers

The Scalar Functions of Hypercomplex NumbersAuthor(s): Henry TaberSource: Proceedings of the American Academy of Arts and Sciences, Vol. 41, No. 3 (Jun.,1905), pp. 59-70Published by: American Academy of Arts & SciencesStable URL: http://www.jstor.org/stable/20022039 .

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THE SCALAR FUNCTIONS OF HYPERCOMPLEX NUMBERS.

By Henry Taber.

Presented March 8,1905. Received March 16,1905.

In a recent number of the Transactions of the American Mathematical

Society * I bave extended the quaternion scalar function to hypercomplex

number systems in general, establishing by the aid of this function

certain important theorems in the theory of hypercomplex numbers, t

I fiud that there are two generalizations of the quaternion scalar function.

Thus, denoting (as, throughout this paper) the constants of multiplica tion of any given hypercomplex number system ex, e2, . . . en) by

yijk, for i,j\k =

1, 2, . . . n, ? when we have

?

w = 2 y**e* (v

= i, 2,... n),

and, by n

A = 2 aie?

any number of the system, I employ SA and SA to denote those functions of the coefficients a and of the constants of multiplication defined as follows :

1 n J? x ? 1 " . J1 SA = n 2. 2.?^' ~SA = n 2. 2.a'Tto n i * i J n

i* i s

When n ? 4, and eue2,eye^ are the units of quaternions, these functions

both coincide with the quaternion scalar function as customarily defined.

Thus, if e4 = 1, and el9 e2, e3 are three mutually normal unit vectors,

we have

7w ? ? ? 7to> 7w

? 1= yjv 0* = 1, 2,3 ; y = 1, 2, 3, 4) ;

* Vol. 5 (Oct., 1904), p. 514.

t In the Proc. Lond. Math. Soc. for Dec, 1890, vol. 22, p. 67,1 had previously extended the scalar function of quaternions to matrices in general, employing this

generalization to prove certain theorems of Sylvester's.



60 PROCEEDINGS OF THE AMERICAN ACADEMY.

and, therefore,

S (?i ci + ?2 02 + ?s ?s + ?4 ^4) = aA= S (?i cx + a2 e2 -f ?3 ?3 + ?4 c4)

by the above definition. For hypercomplex number systems in general,

n n n n

2 2 a?'^? 2 2 "<?*? 1 * 1 > 1 *

1 S

that is, ?-4 ^ SA for every number A of the system.* The chief

properties of the two scalar functions, SA and SA, are enumerated in

Theorems /and LI\>elow; and in Theorem III, I give certain properties

of these functions relating to nilpotent numbers.

Throughout this paper I shall denote by R the domain of rationality of any arbitrary aggregate of scalars including the constants yijk of

multiplication of the number system ex, e2, . . . en, and by 3& (R, et) the

hypercomplex domain of rationality constituted by the totality of numbers

n

-4=2,a?-*' 1

of the system for which the a's are rational in i?.f Any such number

A will be termed rational in this hypercomplex domain. Further, I

shall denote the units of the system reciprocal to ex,e2, . . . en by

ex,e2, . . . en, ?

when, if yijk (i,j,k =

1,2, ... n)

are the constants of

multiplication of the latter system, that is, if

n

?i?j =

2 y&** (V> = 1,2, . . . n),

1 ?

we have

yat =

yak (hj> k, =

l,2,...n);t

and I shall write ? w

A=. 2 ???f

* Thus, let n = 3, and let e3 be a modulus of the system,

el2 = 1 e2

? 0> e2 el = el> e22

= e2

We then have

SA = a34-|a2?

whereas,

t See Trans. Am. Math. Soc, 5, 513.

X Encycl. d. Math. Wissensch., 1,163.



TABER. ? SCALAR FUNCTIONS OF HYPERCOMPLEX NUMBERS. 61

to denote the number obtained from the number

n

-4 = 2 a*e*

i *

by replacing, severally, the units of the original system el9 e2, . . . en by the units ex, e2, . . . en, respectively, of the reciprocal system.

Theorem L Let R denote the domain of rationality of any arbitrary

aggregate of scalars including the constants yijk of multiplication of any

given hypercomplex number system ev e2, . . . en. Let

n

? = 2 a*e<

be any number of the system ; and let

i n n ? i w -J1 ?^ =

-2.2.^-^ SA =

~H 2.?<yy* 71 1 ? 1 J n 1 i 1 3

Then both SA and SA are invariant to any linear transformation of the units of the system ; and, if p is any scalar and B any second number of the system,

SpA=PSA, SpA = pj$A,

S(A + B) = SA + SB, S(A? B) = SA + SB, S AB = SB A, SAB = SB A.

If s is a modulus of the system,

Se=l=z Se.

If A is rational in the hypercomplex domain ?& (R, e?), then both SA and SA are rational in R ; and if, moreover, A is idempotent* there

are ~ . independent hypercomplex numbers, rational in ?& (R, e?),

, idemfaciendf . '

. . . . that are . 7 _ . with respect to A, in terms of which every number

idemjacient * ?

, , idemfaciend . _

q/ ?Ae system . to A can be linearly expressed, and there are

* If A2 = A d? 0, A is idempotent. Benjamin Peirce, Am. Journ. Maths., 4, 104.

t If A B ? B, B is idemfaciend with respect to A ; if B A = B} B is idemfacient with respect to A Peirce, loc. cit., p. 104.




n ? A\ ^> c\ independent hypercomplex numbers, rational in 3? (R, e?),

mlfaciend that are . . with respect to A, in terms of which any number of the

miTacienz

niliaciend

system . . with respect to A can be linearly expressed. Lf A is

n%tjac tenz

nilpotent,\

SAp = 0, SAp = 0

for any positive integer p ; and, conversely, if either S Ap = 0 for every positive integer p, or S Ap = 0 for every positive integer p, A is

nilpotent. Finally, if the system contains n independent numbers

AX,A2, . . .

An for which

SA1 = S A2

= . . . = SAn

= 0,

SA1 =

SA2=. . . =

SAn =

0,

the system is nilpotent.%

The proof of Theorem L, so far as it relates to SA, the first of the two

scalar functions, I have given in the paper above referred to.? This

theorem may be demonstrated for the second scalar function, SA, by the

aid of the following theorems relating to the reciprocal systems e1, e2, . . .

en and el9 e2, . . . en.

Theorem (1). SA = SA, S ? = S A.

Theorem (2). If

e'h = rhi ei + rh2 e2 + . . . + Thn en (h =

1, 2, . . .

n),

and

~e\ ?

r?l ex + rh2~e2 + . . . + rhn ~en (h = 1, 2,

. . . n),

* If A B = 0, B is nilfaciend with respect to A ; if B A == 0, B is nilfacient with

respect to J.. Peirce, loc. cit., p. 104.

t If Am = 0, for some positive integer m, A is nilpotent. Peirce, loc. cit., p. 104.

X A system is nilpotent which contains no idempotent number. Peirce, loc. cit.,

p. 115.

? Loc. cit., p. 514 et seq., and p. 531.




then the systems efl9 e'2, . . . efn and e\, er2, . . .

efn are reciprocal pro

vided the determinant of the transformation is not zero.*

Theorem (3). If p is any scalar, and p A == C, then p A = C; and

conversely. If A + B ==_ (7, then A + i? = (7; and conversely. If A B = (7, then ?A = (7;f and conversely.

Theorem (4). If s is a modulus of the system e1? e2, em then g is

a modulus of the reciprocal^system ex, e2, . . . en; and conversely. If

A is idempotent, so also is A, and conversely ; if A is nilpotent, so also

is A, and conversely. If B is either idemfaciend, or idemfacient, with

respect to A, then B is either idemfacient, or idemfaciend, respectively, with respect to A ; and

conversely._ And, if B is either nilfaciend, or

nilfacient, with respect to A, then B is either nilfacient, or nilfaciend, respectively, with respect to A.%

* For let n

e'i e'j =

%k y'tjk e'k (hj = 1, 2, . . .

w),

n

?j~e\ =

2* 7^ ?4 (t,J = 1, 2, . . . n).

Then n n n

2? 2* ** t^. ??H =

Sa t? 7^ (*',./, / = 1,2, . . . n),

il i

n n n

2* tk rjk r.h ym =

%k tk f,a (ij,/ = 1,2,. . . n) ;

il i

therefore, n n n

%k Tki Wijk ~

y'jJ =

2* 2* Tft r.fc (7^ -

ym) = 0 (i,j, Z = 1, 2, . . . n).

i il

Whence follows

7/?k =

7\jk (*,./,* = 1,2,. . .n),

since, otherwise, the determinant of the transformation is zero.

t If n n n

A B = SZ a,- e,-

* St- &,- ?,-

= 2? cf. ?f.

= (7,

ill

then n n n n

^.^. a.b.y... = S. S. ? 6. 7 -, = c. (& = 1,2. . . . wl : 11 11

and, therefore, _ _ n n n _

BA = ^.b.l?.'^a.7.=:^.c.'e.= C. ill

% This theorem follows from the last clause of (3).




Theorem (5). If the system e1} e2, . . . en is nilpotent so also is the

reciprocal system el9 e2, . . . en ; and conversely.* Theorem (6). If A is rational in 3ft (R, e?), then A is rational in

3ft (R, et) ; and conversely. _ Theorem (7). If Al9 A2, . . .

A^ are independent, so also are Alf

A2, . . .

Ap ; and conversely.!

Assuming Theorem I, in so far as it relates to the first scalar function,

SA, we are now in position to establish the theorem for SA.

First, let n

e\ = 2> Thy ej (?

= 1, 2, . . . ?), i

n

e'h = 2i Tv # (?

= 1, 2, . . . w) ;

i and let

e'i e'j =

2* TV ?'* ft i = 1, 2, ... 72),

i n

"e'i ~?'j =

2* 7 </* ?* ft i = !> 2> *)

Then, by (2),

y'tf* =

Yjik ft? ? = 1, 2, . . . n).

If n ra

^' = 2*a'< ^ =

2*'a< *? =^

i i then

2* ?'.- Ty = ?i 0* = 1, 2, . . . ?) ;

i '

and, therefore, ? n w __

-4' = 2* a'* *?'= 2*

a< ~&i ~ A

i i

But, by Theorem I, SA is invariant to any linear transformation of

the units of the system eu e2, . . . en; that is,

n n

2? 2* ^^ - 2* 2>^ ?#

* For, if ex, e2, . . .

en contains an idempotent number A, then ex, ~?2> ?*

contains an idempotent number A by (4).

t This follows from the first two clauses of (3).




Whence follows ? n n n

2' 2* a'? Yus=22' 2a< ^; 11 * *

that is, $^4 is invariant to any linear transformation of the units of the

system el9 e2, . . . en.

Let p be any scalar, and A and B any two numbers of the system

el9 eL, . . . en. Then, by (1), (3), and Theorem L,

S(pA) = S (pi) =pSl = pSA9

S(A + B) = S(l+?) = Sl+SB=SA + SB*

S (AB) = S (?l) = S (I?) = S(BA).1f

If c is a modulus of the system el9 e2, . . . en, then by (4), s is a modu

lus of the syst?me!, e29 . . . en; and therefore, by (1) and Theorem L,

Se= S~e= 1.

Let A be rational in 3ft (R, ez). Then by (6), A is rational in 3ft (R, e?) ; therefore by Theorem L, SA, and thus, by (1),

S A = SA is rational in R. Let, moreover, A be idempotent. Then,

by (4), A is idempotent. Therefore, by Theorem L, there are

* The equations Sp A = p SA and S (A + B) = SA -\- SB are immediate

consequences of the definition of SA.

t Since eh ei e ?

eh e{ e. we have

n n

2* 7?* 7t?I =

2* 7hik ykjl (i,j, ?, / = 1, 2, . . . n) ; i i

and, therefore, _ -i n n n n

SAB = -$i%j%k

|?

a. b. y.jk yhkh

-.n n n n

W 1 11 1

-. n n n n

SBA = -$, 2 2* S*

? 6,.7iit 7hkh 71 1 1 1 1

-i n n n n

= ? ̂ ' ^/ ^* ^najb?7h?k7w "

i l l i

"Whence follows & J. B ? S B A, since the two last members of these two equa tions are equal, as may be seen by the interchange in either of i and j, and of h

and Jc.

VOL. XLI. ? 5




p. ?

n SA > 0 independent hypercomplex numbers Al9 A2, . . .

A^,

rational in 3ft (R, ez), that are idemfaciend with respect to A; and in terms of these any number idemfaciend to A can be expressed linearly.

By (4), (6), and (7), Ax, A2, . . . A^ are independent, rational with

respect to 3ft (R, ez), and idemfacient with respect to A. Moreover,

there is no number -4^-j-i, independent of Ax, A2, . . .

A^, that is,

idemfacient with respect to A. For, otherwise, by (4) and (7), there are //, + 1 independent numbers of the system idemfaciend to -4, which

is contrary to Theorem L Whence it follows that there are

n SA =znSl= /x, > 0

independent numbers, rational in 3ft (R, e,), that are idemfacient to A, in terms of which all numbers of the system idemfacient to A can be

expressed linearly. Similarly, we may show that there are n

(1 ?

SA)

independent numbers of the system, rational with respect to 3ft (R, e), that are nilfacient with respect to A, in terms of which all numbers nil

facient with respect to A can be expressed linearly.

If A is nilpotent, then by (4), A is nil potent, when by Theorem I, S Ap = 0 for any positive integer p\ and, therefore, by (1) and (3), S Ap = S Ap = 0. If, conversely, S AP = 0 for every positive integer p, then by (1) and (3), SAP =

jSAp = 0 for every positive integer p ;

and, therefore, by Theorem L, A is nilpotent, in which case by (4) A is

nilpotent.

If there are n independent numbers Al9 A2, -4? of the system

el9 e29 . . . en such that

S Ai = SA2

= . . . = SAn

= 0,

then, by (1) and (7), there are n independent numbers A1} A29 . . .

An

of the reciprocal system ex, e2, . . . en such that

Sl? =

Sl2= . . . =

Sln = Q;

and, therefore, by Theorem L, the system ex, e29 . . . en is nilpotent, in

which case, by (5), eu e2, . . . en is nilpotent.

For a system of n = m2 units zi5 (i,j =

1, 2, . . .

n) whose multi

plication table is given by the equations

?#ejk = e,ik, e?jehk

= 0 (i,j,k, h = 1, 2, ... n ; h $j),




the term " quadrate

" has been suggested by Clifford.* A quadrate with

n = m2 units may be termed as a quadrate of order m. The units of a

matrix of order m constitute a quadrate of order m. If we denote by

stj the units of the reciprocal system, we have

*tf =

** (hj =1,2,... m).

A quadrate is thus equivalent to its reciprocal. For the units of a

quadrate we have

Set? = - =

Se?- (i =

1, 2, . . . n), m

Se?J = 0 =

?fy (?,y = 1, 2, . . . n ; i $j) ; f

m m

and, thus, for any number ^. 2-ay c# ?^ sucn a system we have

i i

m m m -t m

s^. = 2/ 2,-% ^

= 2.-??'Se*

? - 2,^? ?

ii i m i

_ m. m _ ot. _ i m SA =

?i? 2i^s** =

2*a*? **=? 2>a<< ii i i

Therefore,

Theorem IT. If eij9 for ij =

1, 2, . . . ?ra, are ?Ae n = m2 units of a

quadrate of order m, that is, if

e*3ejk = ?*> *y*kk

= ? ft./> k,h =

l,29...m;h -^j),

then, if

. A= 22y^% 1 1

is aray number rf the quadrate,

* Peirce, loc. cit., p. 217.

t For, for 1 ̂ i <J m, e . is idempotent ; there are just m independent numbers

idemfaciend to e?, namely, t{. (j' = 1, 2, . . . m) ; and just m independent numbers

e. (j = 1, 2, . . . m) idemfacient to e^. Moreover, for l^i^m, 1 <?,j ^ m,

? =j= J, ?y- is nilpotent.




? l m.

SA=z-~yat,= SA*

Let

Q(A) =

A?+p1A?>-1+ . . . + pm_2A2+pm_1A

= 0

be the equation of lowest order in the first and higher powers of

n

1

and let the roots of the scalar equation

n(x)=X +p1Xm-l+ . . .

+pm_2X+^m_!X = 0

be 0, Xi, x2, . . . xr, respectively of multiplicity v, v1, v2, . . . vr.

Either Am ? 0, or pt, p2, etc., are not all zero, in which case r ^> 0.

For Am =f: 0, and thus r > 0, if we put

t. _ /XV - X^yi fxV

- X2V\V* [XV

- Xrv\>r

I have shown in the paper above referred to that the number ?(A) of

the system is idempotent. t Let the polynomial ? (x) be defined by the

equation

x?(x) = ?(x)f(x).

Then ? (x) is linearly in x2v~l, x*v, etc. Therefore, @ (-4) is a number

of the system. Further, since f (A) is idempotent,

A?(A) = t(A)t(A)=t(A).

* In my first extension of the quarternion scalar function to matrices in general, referred to in the note p. 59, I defined the scalar function SA of any number

m m

? = 2t.2,aje<.. 1 1

of the matrix, or quadrate of order m, as above, namely, as equal to

t Loc. cit., p. 624.




That is, if A is any non-nilpotent number of the system, there is number

? (A),

linear in powers of A, whose product as

pre-factor, or post

factor into A, is equal to an idempotent number.

Let now

SA e1 = SA e2 = . . . = SA en = 0.

Then, for any number

1

of the system, we have

SAB= y.b?SAe?=0.

Therefore, in particular, if A is not nilpotent, we have

SApt(A) = 0

for any positive integer p. Whence follows

S(t(?)f(A)y=St(?) = 0,

which is impossible, since f (-4) is idempotent ; and hence it follows that A is nilpotent. Similarly, if

SAe1 =

SAe2= . . . = SAen

= 0

we may show that A is nilpotent.

Conversely, let A be nilpotent. Then, either

SAex =

SAe2= . . . =

SAen =

0,

or for some number

B=%btet 1

of the system SAB=SBA^0; in which case neither A B nor B A is nilpotent. But, by the above theorem, if A B is not nilpotent, there

is a number Cx su(th that A.BCL = AB. C\ is idempotent ; therefore, a

number Bl = B Cx such that A BL is idempotent. And, if B A is not

nilpotent, there is a number C2 such that C2B.A? C2.BA is idem

potent ; therefore a number B2 =

C2B such that B2 A is idempotent.

Similarly, we may show, if A is nilpotent, that, either




SAex =

SAe2 = . . . =

SAen =

09

or there are two numbers Bx and B2 of the system such that A Bx and

B2 A are idempotent.

We have, therefore, the following theorem :

Theorem LIT. Lf the number n

A = 2^'e*' i

of the system ex, e2, . . . en satisfies the equations

SexA =

Se2A= . . . =

SenA =

0,

or the equations

SexA =

Se2A= . . . =

SenA=zO,

A is nilpotent. Conversely, if A is nilpotent, it satisfies both systems of

equations, or there are two numbers Bx and B2 of the system en e2, . . . en

such that both A Bx and B2 A are idempotent.

Clark University, Worcester, Mass.



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