On the Scalar Functions of Hyper Complex Numbers. Second Paper

On the Scalar Functions of Hyper Complex Numbers. Second PaperAuthor(s): Henry TaberSource: Proceedings of the American Academy of Arts and Sciences, Vol. 48, No. 17 (Mar.,1913), pp. 627-667Published by: American Academy of Arts & SciencesStable URL: http://www.jstor.org/stable/20022864 .

Accessed: 25/05/2014 07:39

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

American Academy of Arts & Sciences is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the American Academy of Arts and Sciences.

http://www.jstor.org

This content downloaded from 194.29.185.113 on Sun, 25 May 2014 07:39:29 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=amacad

http://www.jstor.org/stable/20022864?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


ON THE SCALAR FUNCTIONS OF HYPER COMPLEX NUMBERS.

SECOND PAPER.

By Henry Taber.

?1

In this paper I shall denote by 7^, for i, j, k = 1, 2,... ra, the constants of multiplication of a given non-nilpotent hyper complex number system (e\, e2,. .em).1 We then have

m

(1) eiej =

Z yijhek (h 3 = 1, 2, ... m).

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

(2) A = aiei + a2 d+ ... + amem

of the system (ei, e2,... em) constituting generalizations of the scalar function of quaternions, to which they reduce, becoming identical

when ra = 4, and, at the same time, the system (eh e2, ez, e?) is equiva

lent 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:

1 m m

(3) S^=-Z Z TO, m *=i y=i

-j mm

(4) s2a = - z Z aiyjij>

m -! y-i

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

m l A number A =

? a?ei of any hyper complex system (eu e2, ... em) is ?=l

idempotent if A2 = A 5^ 0; A is nilpotent, if A 5? 0 but A.P = 0 for some positive integer p > 1. A system is nilpotent, if it contains no idempotent number;

otherwise, non-nilpotent. 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).



628 PROCEEDINGS OF THE AMERICAN ACADEMY.

invariant nilpotent sub system of (eh e2,... em), and a method of

determining the maximum invariant nilpotent sub system, if any exist.2 These results are embodied in theorem II.

Theorem I. Let 7^, for i, j, k = 1, 2,.. .m, be the constants of

multiplication of any given hyper complex number system (e-u e2, . . . em).

Let A =

aiei + a2e2 + . . . + amcm

be any number of the system; and let

1 m m

Si 4 = ~ L L Oi7w,

171 t = l 3 = 1

-, m m

S2A = - ? Z ?iT/ij.

m ?=1 y=i

Then both Si A and S2A are invariant to any linear transformation of the system: that is, if

e'i =

Tiie-i + Ti2e2 + . . . + rimem (i =

1, 2, . . .

m),

the determinant of transformation not being zero, a?id if m

e'ie'j =

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

?=i

and

A = ai^i + a2e2 + . . . + amem =

a\e\ + a,2e\ + . . . + a'mefm,

then -, m m

Si A = - ? ? aWV 771 ?=1 ; = 1

-j m w

S2? = - ? ? a'iY'jij. m ?=1 i=i

2 A sub system 2?i, B2, . . . Bp of any hyper complex number system

(61, 62, ... em) is said to be invariant if the product in either order of each

number of (eu e2, .. .em) and each number of (2?i, J32, ... ??p) belongs to the

sub system, for wrhich the necessary and sufficient conditions are

ei Bj =

g'nj Bi + g'2ij B2 + . . . + g'pij Bp,

Bjei =

g"iijBi -f gn2ijB2 -f- . .. + g"VijBp (i

= 1,2, ... w; j

= 1,2, ... p).

An invariant sub system (2?x, i?2, ... 5P) 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 (ei, e2, ... em).



TABER.? SCALAR FUNCTIONS OF HYPER COMPLEX NUMBERS. 629

If p is any scalar, and

B = he! + b2e2+ ... + bmem

any second number of the system, we have

SipA =

pSiA, S2pA = pS2^4,

S,(A ??) = Sx A =*= Sx B, S2(A ? B)

= S2? =*= S2B,

SiAB = SiB?, S2AB = S2??L

7/ e is a modulus of the system,

Si 6 = 1 = S2 .

7/ .4 ?s nilpotent,

S,Ap = 0, S2.4* = 0,

/or guen/ positive integer p; and conversely, if either

SxA* = 0 (p = 1, 2, ... m)

or

S2^ = 0 (p= 1,2, ...m),

^ ?5 nilpotent. Moreover, A is nilpotent if

SiAei =

Si^4e2 = ... =

8\Aem =

0,

or

S2Aex =

S2Ae2 = ... =

S2Aem = 0.

// A is idempotent, there are m Si A > 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 mS2A > 0 linearly independent numbers

satisfying the equation XA = X,

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

Let

(5) X = xid + #202 + ... + xmem,

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 (eh e2. . . em) contain at least one number

satisfying the system of equations

(6) SiXei =

XiSieiCi + x2Sie2ei + ... + xmSiemet = 0

(i=l,2, ...

m).

The resultant of this system being the determinant

(7) Ai = | Si ei ei, Si e2 ei, Siemei

Si 0102, Si0202, . - . Si0m02

Si0i0m, Si02 0m, ... Si0TO0TO

we then have Ai = o. Let X = B be any solution of equations (6).

Then, by theorem I, B is nilpotent. Moreover, for any number A of

(?i, 02,.. .0m), both BA and AB are also solutions of equations (6).

For, for any number

Y = yici + y2e2 + ... + ymem

of (01, 02, ... 0m), we now have

SiBY=yiSiBei + y2SiBe2+ ... +ymS1Bem = 0;

in particular,

Si(BA-ed =

SxiB-Aei) = 0,

Si (?B-*,) =

SiU-Bei) =

Si(Bei-il) =

iSi(B-M) = 0

' (i = 1, 2, ... m).

Since both 2L4 and AB are solutions of equations (6), they are both

nilpotent.

Further, since, for 1 5? i ̂ m, J5 0? is nilpotent, it follows from

theorem I that S2J50? = 0, and thus any solution B of the system of

equations (6) is also a solution of the system of equations

(8) S2Xe{ = ?iS20i0t + x2S2e2et + ... + xmS2emei

= 0

(i =

1, 2, . .. m),

of which the resultant is

(9) ?2 = I S20i0i, S202 0i, ... S20TO0i

S20i02, S2e2e2, ... S20m02

S20i0m, S202 0m, . . . S2emem



TABER.? SCALAR FUNCTIONS OF HYPER COMPLEX NUMBEES. 631

By theorem I every solution of equations (8) is nilpotent. Let Bf be any solution of this system of equations. Precisely as above,

we may show that Bf is nilpotent, and that both BfA and ABf are also solutions of these equations for any number A of the system (ei, e2, . ..

em) ; and, therefore, both B'A and A Bf are nilpotent. Since,

in particular, for 1 ̂ i ̂ m, B'ei is nilpotent, it follows from theorem I that B' is a solution of the system of equations (6).

Let now the nullity 4 of the determinant Ai be m!, where 0<ra'<ra.

There is then a set of just mf linearly independent numbers, Bi, B2 . . . Bm' of the system (e\, e2. .. em) satisfying equations (6) ;

therefore, just m! linearly independent numbers satisfying equations (8), whence it follows that the nullity of A2 is ml\ For 1 ̂ j ^ ra', the product of Bj in either order with any number A of the system is a solution of equations (6) and, therefore, both BjA and ABj are

expressible linearly in terms of Bh B2 . .. Bm'; otherwise, there is a set of more than m! linearly independent solutions of equations (6)

which is contrary to supposition. Moreover, since

S\ (pi Bi + p2B2 + ... + Pmf Bm') ei =

piS?B1ei + piS1B2ei + ... + p^SiB^ei = 0

(i =

1,2, ... ra),

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

.em).

Further, the sub system (B\, B2 . . . Bm') contains every invariant

nilpotent sub system of (eh e2 . . . em), and is therefore the maximum

invariant nilpotent sub system of the latter. For, let (d, C2 . .. Cv)

be any invariant nilpotent sub system of (eh e2 ... em). Since every

number of this sub system is nilpotent, in particular,

S^O (j=l,2,...p). Moreover, since

Qe? =

9jii& + 9ji2@2 + . . . + 9jipCp (i

= 1,2, . .. ra; j

= 1,2, ...

p), we have

SiCjCi =

ftaSid + gji2SxC2 + ... + g^Stf, = 0

(i =

1, 2, ... m; j =

1,2, ... p);

4 The nullity of a matrix or determinant of order m is w! if every (mf ?

l)th minor (minor of order m ?

m! + 1) is zero but not every m'th minor (minor of order m ?

m1). Nullity of order m! is equivalent to rank (Rang) m ? m''.




and thus each of the C's is a solution of equations (6). Therefore, each of the C's is inexpressible linearly in terms of Bh B2 . . . Bm>.

Let

(10) Bf =

bjid + bj2e2 + ...+ bjmcm (j = 1, 2, . .. m').

We may take the b's to be rational functions with respect to the domain R (1) of the constituents of Ax (or of A2) which are integral quadratic functions, rational with respect to R (1), of the constants of multiplication of the number system (ei, e2. . .em). If this number

system belongs to the domain Rf, that is, if its constants of multiplica tion lie in the domain R'', the b's may be so chosen as to lie in this do

main. We may take the B's as m' new units of the number system.

Thus let

(11) *'m_m'+; =

Bj (j -

1, 2, ... m'),

and let e\, er2 ... e'm_m' be any m-m' numbers of (e\, e2 . ..

em) which

constitute with the B's a set of m linearly independent numbers. By what has just been said the coefficients of the transformation

(12) e'i = ru ei + T&e2 + ... + rimem (i

= 1, 2, ... m)

of the number system can be taken rational in any domain to which the number system belongs.

If the number system is transformed by the preceding substitution

(12), and if we put

(13)

then, since

A'1 =

Site's

(i,j =

1,2,. _.m)

Si e'i e'j =

? ? Tih Tjk Si eh ek ?=l k=l

(i, j =

1, 2, ... m)

we have

(14) A'i =

T2Ai,

where T is the determinant of the substitution. Similarly, if

(15)

we have

(16)

A'2 =

(i,j =

1, 2,. . .m)

A'2 =

PA2

Therefore, the equations Ai = 0, A2

formation of the units of the svstem. 0 are invariant to any trans




Let now Ax =^ 0, in which case A2 ̂ 0, and there is no number of the

system satisfying equations (6), or equations (8) ; and, therefore, the

system contains no invariant nilpotent sub system. In.this case,

therefore, if

SiAei= SiBei (i = 1, 2, ... m),

we have A ? B; otherwise, A ? B ?? 0 is a solution of equations (6).

Similarly, if

S2Aei= S2Bc? (? = 1,2, ... m),

then A = B. We have now the following theorem.

Theorem II. Let (ei, e2. . .em) be any non-nilpolent hyper complex number system; let

X = xxei + x2e2 + . . . + xmem,

and let Ais Si e? ej

(i,j =

1,2, ... m)

S2 ei ej

(i,j =

1,2, ... m)

be the resultants, respectively, of the two systems of equations

(a) SiXei =

XiSieiei + x2Sie2ei + . . . + xmSiemei = 0

(i =

1, 2, . . . m),

and

(?) S2Xei =

xxS2eiei + x2S2e2e{ + . .. + xmS2eme{ = 0

(i =

1, 2, .. . m).

Then, if the number system is transformed by the substitution

e'i = T?ei + T&e2 + . . . + Timem (i

= 1, 2, . . . m),

and if

A'x =

we have

Si e'i e'j |,

A'2

(i,j = 1,2, ... m) I

A', =

f2Au

S2 e'i e'j

(i, j =

1,2, ... m)

A'2 =

PA2,

where T is the determinant of the substitution. Further, the condition, necessary and sufficient, that the number system shall contain no invariant

nilpotent sub system is that Ai 9^ 0, or A2 9e 0. In this case, if either




SxAe{= SiBei (i =

1, 2, ... m) or

S2Aei =

S2Bei (i = 1, 2, ... m),

we have A ? B. If Ai =

0, then A2 =

0, and conversely; moreover,

the nullity of Ai is equal to the nullity of A2. Every number of the

system satisfying equations (a) is a solution of equations (?), and con

versely. If B is any solution of equations (a) (or of equations (?)), then, for any number A of the system (eiy e2 . .. em), both B A and A B are

solutions of these equations. If the nullity of Ai is m!, there is a set

of just mr linearly independent solutions of equations (a) (or equations (?) ); and any such set of mf numbers of (e\, e2, ... em) constitute an

invariant nilpotent sub system containing every invariant nilpotent sub

system of (elf e2, ... em).

Let the system (ei, e2, ... em) contain a nilpotent sub system

(C\, C2, ... Cp) such that

v

Cjei =

L SijhCh (i = 1,2, ... m; j

= 1,2 ... p). A=l

For 1 ̂ j^p, wre then have, by theorem I, p

SxCjei =

? gijhSiCh =0 (i = 1, 2, ... m);

h=i

therefore, Ai =

0, and thus (e\, e2, ... em) contains an invariant

nilpotent sub system to which the sub system (Ci, C2, ... Cp) belongs.

Similarly, we may show that, if the system (e\, e2, ... em) contains a

nilpotent sub system (Ci, C2, ... Cv) such that.

p

eiCj =

? gi?Ch (i = 1,2 ... m; j = 1,2 ... p),

h=l

it then contains an invariant nilpotent sub system which includes the sub system (Ci, C2, ...

Cp). If (e\, e2, ...

em) contains a sub system (Ci, C2, ... Cv)

such that

S\C\ =

S\C2 = ... =

SiCp = 0

or

?>2w =

02C2 =...== S2Cp

= 0,

this sub system is nilpotent, since then, by theorem I, every number of

the sub system is nilpotent. Thus, if

C = g1C1 + g2C2+ ... + gpCp




is any number of the sub system, we have

C" = g??Wi + gttoCt + ... + gfwCp;

therefore,

Si C? = giw S\ Ci + g?*> Si C, + - -. + 9vw Si Cv

= 0,

for any positive integer q.

?2.

For any given number m

A = ? a?e?

*=i

of the non-nilpotent system (ei, e%, ... em) there is a linear relation

between A, A2, ...

ylw+1; therefore, a smallest positive integer

/x ? m + 1 for which A, A2, ... ^4M are linearly related, and thus for which we have

(17) fl (A) = A? + piA^ + ... + p,_iA

= 0,

where the p's are functions of the a's. Let pi, p2, ... pr, respec

tively of multiplicity fxh pL2, ... jur, be the distinct non-zero roots, if

any, of 0 (p) =

0; when we have

(18) ? (p) s pfco (p -

Pl)h (p -

P2)h ... (p- Pr)kP)

where &o = 1. Further, let

(19) W (p) ^ p(p- pi) (p- P2) ... (p -

pr).

Let now

/U) = L cnAh A=l

be any polynomial in A. l? f (A) ?

0, then p ^ ju and/(p) contains

?(p); otherwise, there is a linear relation between A, A2, ...

^4M?*, which is contrary to supposition. Wherefore, if f(A) is nilpotent, / (p) contains W (p). Conversely, if / (p) contains IF (p), / (A) is nil

potent; and, if/(p) contain ? (p), then/(A) = 0.

Let ^4 be non-nilpotent. Corresponding respectively to the r ̂ 1 distinct non-zero roots of ?2 (p)

= 0, are r linearly independent num bers Ii, 12,

... IT, linear in powers of A, which are severally idempo

tent and mutually nilfactorial : thus we have

(20) IJu =

Iu^O, IJV = 0 (u, v = 1, 2, . . . r; v ̂ u).




If, for 1 ? u < r,

(21) -

,,,.,M.(^^::(:-/.>7, f(n_n \ku_(n _n \ku\kv

<t>v{U)(p) = ' ̂

? (pv

? Pu)?u

(v =

1, 2, ... w ? 1, u + 1, ...

r),

and

(22) /? (p) = 0O(?) (p) fc<?) (p) ... ?fc^c?) (p) ^eo (p) ... $,<?> (p),

we may write

(23) h = fu(A) (u= 1,2, ...r).?

I shall denote by r the greatest value of r for any number A of the

system. Then r is the greatest number of idempotent numbers, mu

tually nilfactorial, contained in the system (e\, e2, ... em). For, if

possible, let the system contain p > r numbers K\, K2, ...

Kp satis

fying the conditions

K\ =

Ku * 0, KUKV = 0

(u, v = 1, 2,

.. . p; v t? u).

The K's are then linearly independent. If now

A = X? + \2K2 + ... + \PKP,

where the X's are any p distinct scalars other than zero, the equation

Q, (p) = 0 has p > r distinct non-zero roots, which is contrary to

supposition. Let A be non-nilpotent and, for any positive integer p, let

(24) NW = wp(A) s= A* - Z Pp^;

w=l

5 For then, in the first place, fu (p) contains P as a factor; therefore, fu (A) is linear in powers of A. Moreover, for 1 <u<r, fu(p) does not contain

?(p), whereas (fu(P))2 ?

fu(P) does contain Q(p) ; and, therefore, 7W =

/M(A) ^ 0,

Iu ?

Iu = 0. Further, for any two distinct integers u and t> from 1 to r,

/u (P)fv (P) contains ?2 (p); and. therefore, IUIV = 0. By the aid of the above

two equations, we may show that h, h, . . . Ir are linearly independent. Thus, if

J =

ah + coI2 + . + crIr =

0,

then, for 1 < w < r, cu/u =

IUJIu =

0;

and, therefore, cu = 0.




in which case N(p> is nilpotent, since

r

&P (p) = PP ? Z PuPfu (p)

w=l

contains W (p) : therefore, by theorem I,

(25) SxAv= Z PupSJu+ SiNW= ? PupSJu.

U=l M=l

S*A*= Z p,pSsJ?+&#<?= 21 P?pS2/a. M=l U=l

If possible, let

Si# =

S^1 = ... =

SiA***-1 = 0

for some positive integer p. By (25), we then have

pr?+/iSi/i + P2p+hsj2 + ... + Prhsjr = o

(? =

0,l,2,...r-1);

and since, by theorem I, neither $1/1, S1/2, . nor S\Ir is zero, it fol lows that

Pip, . . Pv+r~l I

= 0,

fh?, . . . PrP+r^

which is impossible, since by supposition the p's are distinct and other than zero. A fortiore, we cannot have

SiAp =

SiAp+1 ==... = Si^^"1

= 0

for any positive integer p. Similarly, we may show that we cannot have

S2Ap =

S2AP+1 = ... =

S2Ap+?~l = 0

for any positive integer p if A is non-nilpotent. We have now the following theorem.

Theorem III. L,et (e?, e2, ... em) be any given non-nilpotent number

system; and let r be the maximum number of idempotent numbers, mutually nilfactorial contained in the system. Then, if for any number

A = aid + a2e2 + ... + amem




of the system, tve have, for some positive integer p,

SxAv+h = 0 (h =

0, 1, 2, ... r - 1), or

S2A*+h = 0 (h

= 0,1,2, ...r-1),

A is nilpotent. Conversely, if A is nilpotent, these equations are alt

satisfied for any positive integer p.6

With respect to the idempotent numbers 7i, I2, ... Ir, linear in

powers of any non-nilpotent number A, the number system may be

regularized as follows. Let roo denote the aggregate of numbers

r r r r

ei? Z hex? Z 6ilu+ Z Z lud?? W=l W=l u=l v=l

for i = 1, 2, ... m. For any assigned integer u from 1 to r, let Tm.

and roM denote, respectively, the aggregates

r r

hei? Z lueilv and erfu ?

Z hedu

for i = 1,2, . .. m; and, for any assigned pair of integers u, v from

1 to r, let Tuv denote the aggregate of numbers Iueilv for i = 1, 2, ... m.

Further, for u and v any two integers from o to r, let raMC denote the

greatest number of linearly independent numbers of the aggregate Tuv;

and, if muv 5? 0, let Juhv, for h = 1,2, ... raM?, denote any system of

muv linearly independent numbers of Tuv. We then have, by (20),

(2o) luJuhv == Juhv

= JuhvJ-v, JLuJuh'o

== Juh'o J oh"vJ-v

= J oh"v

(u,v= 1,2, ... r; h= 1, 2,... muv; A'= 1,2,. ..muo; h" =

1,2,... m0v),

(27) -l u'Juhv == U =

JuhvJ-v'

(u, v = 0,1,2,

... r; h = 1,2, ... rawt); t?V =

1,2, ... r; u ?? u,v' 5? v)..

We may now show that the J's are linearly independent. For, if

J = Z Z Z ?phqJphq + Z Z 9'phoJpho 2>=1 fl=l ? = l P = l ? = l

r wiop Woo

+ Z Z 9ohpJohp+ Z $ oho J oho = 0^ p=l /i=l A=l

6 Cf. paper by the author in the Trans. Am. Math. Soc, 5, 545, note.




then, for any pair of integers u, v from 1 to r^

muv

/ t guhvJuhv ~

J-uJ Iv =

v); h = l

and, since by supposition Juw, Ju2v, etc., are linearly independent, we

have

guhv =0 (u,v=l,2,...r;h=l,2,... muv).

Whence it follows that

r mpo r >op moo

J ? 2- 2- gphoJpho T L jL gohpJ ohp~T 2w g oho J oho =

0; p=l h = l p=l h = l h=l

and, therefore, for 1 ? u ^ r, >uo mou

/, guhoJuho =

iiie/ =

U, 2^ gohu*J ohu == J lu == u.

A=l ?=l

From these equations we derive

#wAo = 0 (m

= 1, 2, ... r; h =

1, 2, ... mM0);

#oAu =0 (u = 1,2, ... r; h =

1,2, ... mou).

Thus, ultimately, we have m00

J ? IL g oho J oho = 0;

? = l

whence follows

g0ho = 0 (h = 1, 2, ... m00).

Since

r r

(28) e,-= ? Z ^?7, M=l 0=1

r r r r

+ Z (^tt?i ? Z i?^ ?) + Z (eiiu ?

Z ^^/u) w=l v = l u=l v=l

r r r r

+ (et ? Z lu?i ?

Z etlu+ Z Z heilt) u=l u = l u=\ v=l

(i =

1, 2, ... m),

it follows that each unit of (ei, e2, ... em), and thus that any number of this system, can be expressed linearly in terms of numbers in the

(r + l)2 aggregates Tuv (u, v = 0, 1, 2, ... r), and, therefore, linearly




in terms of the J's. Whence it follows that we may take the J's as new units, and the number system thus transformed is regularized with respect to the idempotent numbers 1\, I2, ...

IT.7

Since, for 1 5? u ^ r, Iu belongs to Tuu, we may put

(29) Iu = Jumuuu (u =

1,2, ... r).

If now B' is any number of the system (e\, e2, ... em) satisfying the

equation Iu B' = B'', then, by (26) and (27), r muv

B' = Z Z b'vhJuhv;

similarly, if B"IU = B", we have

r mvu

B" = Z Z b"vhJvhu v=0 h=l

Therefore, by theorem I. r

mSiIu = Z muv,

(30) mS2Iu =

Z m,,M, (u= 1,2, ...r)'

Let (w, v), for t?, v any two integers from 0 to r denote a number of

the aggregate Tuv. From (26) and (27), it then follows that the non

vanishing products of numbers in the several aggregates are given by the following equations:

(31) (u, v) (v, w) =

(u, w)

(u, v, w = 0, 1, 2, . . .

r);

and wre further have

(32) (u, v) (vf, w) = 0

(u, v, v', w = 0, 1, 2, ... r; vf ?? v).8

7 When the number system is thus transformed each of the new units is

in one or other of Peirce's four "groups" or aggregates with respect to each

of the r idempotent numbers Iu h, . Ir- Thus, if u is any integer from

1 to r and v, w any two integers from 0 to r other than u, then the units

Juh]u (1 ^ hi < muu), Juhn (1 $ h2 < muv), JvhsU (1 ^h% < mvu), and

Jvh w (1 S h4 < m)m are respectively in the first, second, third, and fourth

groups with respect to Iu. See B. Peirce, loc. cit., p. 109.

We have now r r

m = 2 2 >u* M=0l) = 0

8 Cf. B. Peirce, loc. cit., p. 111.



TABER.? SCALAR FUNCTIONS OF HYPER COMPLEX NUMBERS 641

Therefore, if in the square array,

Tn, Ti2, ... Tir, Tio

I^i, T22f ... T2r, T2Q

Tn, Tr2f . . . Trr, Ttq

Foi, To2, ... Tor, Too

we strike out any p rows or any p columns, the units of the aggregates

in the resulting array constitute a sub system of (eh e<?, ... em). In

particular, for 0 ^ u = r, the units of Tuu constitute a sub system. Since, by (32), (u, v) is nilpotent if u 5? v, we have

(33) S^u, v) =

0, S2(u, v) = 0 (u, u = 0, 1, 2, ... r; v ̂ u).

Let now A be so chosen that r = r, where, as above, r is the greatest

value of r for any number A of the system. The units of Too then constitute a nilpotent sub system; and, since every number of a

nilpotent system or sub system is nilpotent, we now have

(34) Si (0,0) = 0, S2(0 0) = 0.

For, otherwise, if Too contains an idempotent number J0, we have

70 Iu = 0 =

Iu I0 (u= 1,2, ... r)

by (27) ; and thus the number system (e\, e2 .. . em) contains r + 1

idempotent numbers mutually nilfactorial, which is impossible, as shown above p. 19. Moreover, for 1 ? u = r, there is now but one

idempotent number in the aggregate Tuu. For, if possible, let Tuu muu

contain a second idempotent number I'u = ? Ch Juhu other than Iu,

?=l

in which case we have I'u2 =

Vu\ let

I" = I ? V

when we have, by (20) and (26),

Jtf 2 ? J 2 _ J jr Jt J i 77 2 _ 7 O 7' I 7' _ J" * U ?U J-U L U 1 U 1U I * U ?

LU ?-L U \ *- U ? ? Uf

*- uL u ? ukJ-u L u) u \Lu * u) * u ? * U? U)

and, by (32),

IJ"U =

1,(1, -

/'?) = 0 =

(/, -

/'?)/? =

/",/, (c * u). 9 Cf. B. Peirce, loc. cit., p. 112.




Wherefore, there are then at least r -\- 1 idempotent numbers mutually

nilfactorial, namely, Fw Inu and Iv for v ? 1,2, . . . u ?l,u-\-l, . . . r,

which is impossible. The number system when regularized with respect to r idempotent

numbers, so that r0o contains no idempotent number, and each of the

aggregates Tu, T22, . . . Trr but a single idempotent number, is said to

be completely regularized. For 1 ~ u ~

r, we may now take the muu ? 1 units other than Iu

of the aggregate or system Tuu so that they shall all be nilpotent; in which case they constitute by themselves a nilpotent sub system, every number of which is, therefore, nilpotent.10 I shall assume that in each of the aggregates Tuu (u

? 1, 2, . . .

r) the units have been so

chosen.

Let r r muv

(35) A = Z Z Z auhvJuhv. w=0 v=0 h = l

By equations (29), (30), (33), and (34), and by what has just been

stated, we now have r

(36) Si A = 21 Clumuu^Slh w=l

1 r f =

?^ 2^ aumuuumuvj m u=l v=0

r

(37) S2A = 2- #wmUUM*S2iu

w=l

1 r r ? ~

Z Z #uroMUtt7ttCM.

I shall say that the two idempotent units Iu and Iv (1 = u ? r, l^v = r, v t? u) are connected if there are two numbers (u, v)f and

(v, u)' such that

Si (u, v)f (v, u)f ?? 0;

otherwise, not connected. If Iu and /? are not connected, then

Si (u, v) (v, u) = 0

10 This theorem is due to B. Peirce, loc cit., p. 118. His proof is defective. The first proof, I believe, of the theorem without the aid of the theory of

groups was given by me in the Transactions American Mathematical Society, 5, p. 547, by employing the generalized scalar function.



TABER,? SCALAR FUNCTIONS OF HYPER COMPLEX NUMBERS. 643

for any two numbers (u, v) of rM? and (v, u) of Tvu. Let (u, v)\ (v, u)' be any two numbers of Tuv, Tm respectively. Then

(u,v)r (v,u)f =

plu -

Nu

by (31), where Nu is linear in the nilpotent units of Tuu and is, there

fore, either zero or nilpotent, and thus Nup+1 = 0 for some positive

integer p. Furthermore,

Si(u,vY (v,u)f =

pSihu

If now h and Iv are connected, then, for a proper choice of (u, v)f, (v, u)', we have Si(u, v)f (v, u)' t? 0, in which case p ^ 0: therefore,

we may put

(Vj u)" =

-|ri fo u)' (PVI- + PP~1N- + + P^'1 + N?P}>

when we have

(u, v)' (v, u)" =

-?j (plu -

Nu) (pp Iu + Pp~lNu + ... + pNfi + Nup) r

= _1_ (nP+l T _ AT P+l) ? T . ?

p+1 \P J-U J-l U J ?

?U,

and since

[(*, u)" (u, v)J =

(v, u)". (u, v)f (v, u)". (u, vY =

(v,u)"Iu(u,v)' =

(v,u)r,(u,vY, it follows that

0, u)" (u, v)! =

Iv,

otherwise, there is more than one idempotent unit in Tvv, which is contrary to supposition. Wherefore, if Iu and Iv are connected, there are two numbers (u, v)' and (v, u)', of Tuv and Tvu respectively, such that

(u, vY (v, u)f =

I?, (v, u)r (u, v)r = Iv;

and conversely, since in this case

Si(u,v)'(v,u)f =

SJu^O.

If Iu and Iv are connected, and Iv and Iw are also connected, then Iu and Iw are connected, where u, v, to are any three distinct integers from

11 Further, S2 (u, v)' (v, u)' =

pS2Iu) therefore, if Si (u, v)' (v, u)r ?? 0, then S2 (u, v)' (v, u)' ^0, and conversely.




1 to r. For, in this case there are two pairs of numbers, namely,

(u, vY, (v, u)f and (v, w)f, (w, v)f such that

(u, vY (v, u)' =

Iu, (v, u)' (u, v)f =

Iv,

(v, w)' (w, v)r =

Iv, (w, v)' (v, w)' =

Iw.

Therefore, if

(u, w)' =

(u, v)' (v, w)r, (w, u)f =

(w, v)f (v, u)f,

we have, by (26),

(u, w)f (w, u)f =

(u, v)''. (v, w)' (tv, v)'. (v, u)f =

(u, v)' Iv (v, u)f =

(u, v)' (v, u)' =

Iu,

(w, u)' (u, w)' =

(w, v)'. (v, u)f (u, v)f. (v, w)' =

(w, v)' Iv (v, w)' =

(w, v)' (v, w)f =

Iw.

For u, v any two distinct integers from 1 to r, let Iu and Iv be con nected. Thus let

(u, v)' (v, u)' =

Iu, (v, u)f (u, v); =

Iv.

Let k = muu ? 1 ; and let the nilpotent units of Tuu be denoted by

Nu(l\ Nu{2\ . . N . Then (u, v)' and the products Nu{h)-(u, v)', for h =

1, 2, . . . k, are numbers of the aggregate Tuv linearly inde

pendent. For, if k

9o '

(u, v)f + Z ShNuW .

(Uj vy = o,

h=i

then u k k

golu + Z 9hNu^ =

[go (u, v)' + ? QhNuM (u, v)'] (v, u)' = 0

A=l A=l

which is impossible, unless the g's are all zero. Therefore,

muv ̂ k + 1 = mm>

Moreover, there is no number in the aggregate Tuv linearly indepen dent of these k + 1 numbers of this aggregate. For, if (u, v) is any number of this aggregate, since (u v) (v, u)r belongs to the aggregate Tuu, we have

k

(u, v) (v, u)' = c0Iu+ L chN?h);




and, therefore,

(u, v) =

(u, v) Iv = (u, v) (v, u)' (u, v)' k

= (colu + ? ehNuW) (u, vY

A=l

k

= c0(u,vY+ ? chNu^(u,v)\

A=l

Whence it follows that muv cannot exceed muu = k + 1; and, there

fore, muv =

muu. Similarly, we may show next that (v, u)f and the

product (v,u)'N?h\ for h = 1,2, ... k, are linearly independent,

and that in terms of these numbers every number of the aggregate Tw can be expressed linearly. Finally, that /? and the k products (v, u)''Nuh^(u, v)', for h =

1, 2, ... k, are linearly independent, and

that in terms of these numbers every number of the aggregate Tn can

be expressed linearly. Therefore, in particular, if Iu and Iv are con

nected,

U^uu :==

'"/uv ""vu nZw

For l^i = m and u, v any two integers from 0 to r, let (u, v){ denote the component of et in Tuv. We then have

(38) ei= ? ? (p,q)i (i= 1,2, ...m). p=0s=0

Whence, from (32), we derive

(39) Si (u, v) ei = ? ? Sx (u, v) (p, q)i

p=0a=0

r

= ? Si (u, v) (v, q)i

3=0

r

= ? Si (v, q)i (u, v)

= Si (v, u)i (u, v) = Si (?, v) (v, u)i

q=0

(u, v = 0, 1, 2 .. .r; i =

1, 2, ... m).

We may now show first that if, for 0 ^ u ? r, the aggregate Tq* contains any unit, that is, if mu0 > 0, the number system (ei, e2, ..

.em) contains an invariant nilpotent sub system. For, let (u, o) ?? 0, and let

(o, u)i(u, o) =

(o, o)/ (i =

1,2, ...m);




when, by (34) and (39), we have

Si(u, o) ei = Si(u, o) (o, u)i

= Sx(o, u)i (u, o)

= Si(o, o)i

= 0

(i =

1,2, ... ra),

and thus (u, o) satisfies equations (6). Similarly, if mou > 0(1 ^u^.r), we may show that (eh e2, ...

em) contains an invariant nilpotent sub

system.

Again, if Tuu(1 tku^r) contains more than one unit, that is, if

muu > 1, the system (eh e2, ... em) contains an invariant nilpotent sub

system. For, in this case, there is a nilpotent number (u, u) of Tuu whose product with any number of this aggregate is, therefore, nil

potent;12 and thus (u u) (u,u)i} for i = 1, 2, .. . ra, is nilpotent: therefore,

Si(u, u)c? =

S\(u, u) (u, u)i = 0 (i

= 1, 2, . .. m),

and thus (u, u) is a solution of equations (6). If, for u, v any two distinct integers from 1 to r, Iu and Iv are connected, and either Tu9 or Tvu contains more than one unit; that is, if either muv > 1 or

mm > 1, the system (eh e2, ... em) contains a nilpotent sub system.

For then, by the theorem p. 645, we have muu > 1. Further, if Iu and

Iv are not connected, and either Tuv or Tvu contains one or more

units, that is, if muv > 0 or mm > 0, the number system contains an

invariant nilpotent sub system. For let (u, v) ^ 0: in this case, by the theorem given, p. 642, we have

Si(u, v) (v, u)i =0 (i = 1, 2, . . . ra);

therefore,

Si (m, v)ei =

Si(u, v) (v, u)i = 0 (i

= 1,2, ... ra),

and thus (u, v) satisfies equations (6). Finally, if Iu and Iv are not

connected and mvu > 0, (e\, e2, ... em) contains an invariant nilpotent

sub system.

12 Namely, when muu > 1, any number (i?, u) linear in the nilpotent units of Tuu is such a number. For since Iu is a modulus of the system Tuu, these

nilpotent units constitute an invariant nilpotent sub system of Tuu- Where

fore, the products of (u, u) and any number of Tuu belongs to this nilpotent sub system, and is, therefore, nilpotent.




I shall now assume that the number system (e\, e\, ... em) contains

no invariant nilpotent sub system, in which case, by what has just been proved, we have

(40) muo =

mou =

m00 = 0 (u

= 1, 2, ... r),

that is, no number of the system is contained in roo nor in either of the

aggregates TUOf Tou for u = 1, 2, ... r. Further,

(41) muu =1 (u= 1,2,... r),

that is, Iu is the only unit in Tuu for 1 ̂ u ^ r. Finally, for u and v

any two distinct integers from 1 to r, if Iu and Iv are connected,

^uv ==

Wim =

-1;

whereas, if Iu and /? are not connected,

m = mvu

= 0.

In the present case, the number system contains a modulus, viz.,

(42) e = h + I2 + ... + Ir,

since, for u, v any two integers from 1 to r, if Tuv contains a unit Ju\v, we have

? Julv = *1 uiv

==: *^m1?

by (26) and (27). It is, with the present assumption, convenient to modify our nota

tion to indicate the connection which may exist between certain of the idempotent numbers, Il9 I2, ... I~r. I shall, therefore, suppose

these numbers arranged in v aggregates, 1 ^ v ̂ r, containing respec V

tively fjLi, fjL2, ... fiv of the Fs, where Z Mp =

r> any two idem

potent numbers in the same aggregate being connected, but no

pair of idempotent numbers in different aggregates being connected; and, for 1 =p = v, I shall denote by Iu{p) (u

? 1, 2, . . .

jjlp) the idem

potent numbers in the pth aggregate. The r2 aggregates of numbers,,

formerly denoted by TUjV for u, v = 1, 2, .. . r, into, one or other of

which the units fall when the system is regularized as above and contains no invariant nilpotent sub system, will now be denoted by

Tu}p'q) for p,q= 1,2, ... v, and for u =

1,2, ... ?jlv and v =

1, 2, . . . fiq',




and the number of linearly independent numbers in TuJp'q} will be denoted by mu/p,g)u. By what is shown above we now have

(44) mjp>p) = 1 (v

= i, 2, . . . v; u, v = 1, 2, . . . Mp),

(45) m? > = 0

(p,q= 1,2, ... v; q^ p; u= 1,2, ... ?p; v =

1,2, . . . /?g)14.

For 1 ~ p ? v and u and v any two distinct integers from 1 to ppy

we may now, in harmony with the preceding notation, denote the single unit of Tuv(p,v) by Juv(p} ; and if, further, we denote by Juu(p) the idem

potent unit I?p) of TUup,p\ we shall have as the multiplication table

of the system

(46) Ju?P)Jvw{P) =

PuvwJuw^, Juv{v)J?w{p) = 0

(p =

1, 2, ... v; u, v, v, w = 1, 2, ... up; v ^ v),

(47) Ju?p)J*/q) = 0

(2?, q =

1, 2, ... y; ? ^ p; w,fl =

1, 2, ... /?p; u, v,= 1, 2, ... /?g)

by (31), (32), and (44), where pmv = puvv = 1. For 1 ? p = ?> and for

^, v any two integers from 1 to /xp, it follows from (44) that

and thus puJp) j? 0, otherwise Jm{p) =

7u(p) and JJp} =

7^) are not

connected; and, since

(pWu^)2Juu^= (puJP)JuuiP))2= (JJP)Jm^)2 =

Ju?p)>J*u{p)Ju?p)-J^p) =

p <p)Jwmjnwjf?p)

= Pvu?P)Juv{P)JJP)

= Pvu^Puvu{P)Juu^ ?? 0,

we have pw (2?) = Ptu.(p). Further, for 1 ? w ^ /xp,

PufwWpwWJfW =

PuW(P)J^P)e/^P) =

Jw{P)'JuV(P)JvU>{P)

= J,u{P)Ju^'J^P)

= Pvuv{P)JJP)Jvw{P)

= Pvu?p)JJp) * 0;

and, therefore, pUw(p) 5^0.

13 Thus, whereas, formerly i\b denoted the aggregate of numbers /?e^/,

for ? = 1, 2, ... m, of which mM1) were linearly independent, TuP? is now the

aggregate of numbers Iu{p) eiU{q) for t = 1, 2, . . . ra, of which mup^ are

linearly independent. 14: Therefore,

v v Mp My v Pp V-p v

m = S S Z S mUr(p?fl) = 222 mMl,^

= 2 MP2.

p=lg=lli=l?=l p=lu=lu=l p=l




Let

(48) /?,<*> =

?-Jui^Jiv^ vpiulPlvl

(p =

1,2, ... v; u,v =

1,2, ... ?v).

Then

(49) j?<*)=-^ j*p? jlf w = V^yjw(? Puitr

' Pmi?^;

(p =

1, 2, ... v; u, v = 1, 2, ...

fjLP)

by (46) ; and, therefore, we may take the J's as new units. We now

have

(50) JuP?jn = , ? *

? , JulWJu<?>-J?<*>Ji.w ^Plul{P)Plv?P)-piv?P)pUP)

Vpiui^Pl?!^

Ju?P)J*w(P) = , 1

J?fl>WP) ' J??V) Jlw{V) = 0

VPl^W(p)wi(pWp)

(p =

1, 2, ... v; u, v, v, w = 1, 2, ... jup; i/ ^ a),

(51) J>)J^C? =

-^-^-^- j^p; jlf(rt. Ju^)JlvM) = 0 Vplia(|,>Pm(w-pi^iwpWfl)

fe g =

1, 2, ... v; q j? p; u, v = 1, 2, ... pp; u, v =

1,2, ... p,q).

For 1 = p = v, the units Ju}p) for w, ? = 1,2, .. .

jup constitute a quad rate of order ?jlp; and, therefore, in the present case, the number sys

tem is constituted by v mutually nilfactorial quadrates.16 For the

modulus e of the system we now have

V ?p V Pp _

(52) e = I I J,w =11 >UM<P> P=l M=l p=l W = l

15 For

JuitoJvW'JnMJwW =

Jui^-Jiv^Jn^-Jiw^ =

Pm(p)Jui(p)Jn(p)Jiw(p) = Pivl(p)Jui(p)Jiw(p).

16 A quadrate is_a hyper complex number system with m = m? units uv (u, v =

1, 2, . . . m) which can be so chosen that

eUv VW = uw, uv v'w = 0 (u, v, v', w =

1, 2, . . . m; v' ?? v). B. Peirce, Am. Journ. Maths., 4, 217.




By (30), (33), (40), (44), and (45), we now have

v Mg Mp

(53) mSju?v) =

Z Z Wp'9) = Z <uv{p'p) =

?v g=l v =1 0=1

Mp v V-q =

? Wp'p) = E ? W'p> =

mSt~J?<*> 0=1 g=l 0=1

(p =

1, 2, ... v; u = 1, 2, ...

fxp),

(54) Si!v?& = 0, S2JU?V) = 0

(p =

1, 2, ... v; u, v = 1, 2, ... fjLp; v ^ t?).

m

And since, for any number A = Z a*^ we may now Put

*=i

v lip tip __

(55) .4 = I ? I cJ*>J?w, P=l tt=l 0=1

we have

v up fip

(56) SU = ? ? ? cJ*)Sj?*> p=l tt=l 0=1

v Pp lip _ =

? ? ? c?(')S2JMw = M. P=l M=l 0=1

Therefore, in particular,

(57) Stfi?j =

S2eiej (i,j = 1, 2, ... m),

and thus we have Ai =

A2 also in the case now considered, when the

system (elf e2, ... em) contains no invariant nilpotent sub system and

neither Ai nor Ao is zero.

From the conditions, necessary and sufficient, that the m3 constants

Jijk (if j> k = 1,2, ... m) shall constitute the constants of multiplica tion of a hyper complex number system in m units, viz.,

m m

(58) Z yijklkhl = Z 7ikl7jhk k=l ?=l

(i,j,h,l =

1,2, ... m),




we derive

(59) mmA1= | mSifaej)

(i,j =

1,2, ... m)

2h2k7ijk7khh

(i,j =

1,2, ... m)

7ill, . . - 7ilw, . . 7iml, . . . 7imm

(i =1,2, ... m)

(60) wwA2 = | mS2e-3ei

(i,j =

1,2, ... m)

2h2k7jik7hkh

(i,j =

1,2, ... m)

7l?, . . 7lim) 7mily . . 7mim

(i= 1,2, ...m)

= j 2h2k7ikh7jhk

I (h j =

1,2, ... m)

7jll, 7jmh 7jlm, . . . 7jmm

(j= 1,2, ...m)

2h2k7hjk7kih

(i,j =

1,2, ... m)

7ljh 7mjl, . . 7ljm, . . 7mjm

(i= 1,2, ...m)

A number system containing no invariant sub system is termed by Cartan a simple system (syst?me simple), and he shows that such a

system is what is here termed a quadrate. A non-simple system containing no invariant nilpotent sub system Carten terms semi

simple.17 Such a system is constituted by nilfactorial quadrates of which the invariant sub systems are any p (1 ̂ p < v) of these quad rates. By what is shown above it appears that Ai ^ 0 or A2 9a 0 is the condition necessary and sufficient that a number system shall

be either simple or semi-simple. We have, therefore, the following theorem :

Theorem IV. Let eh e2, ..

number system, and let

Ais Sie^ej

(i j = 1, 2, r)

be the units of any hyper complex

S2eiej

(i,j =

1,2, ...

r)

Then Ai =

A2. H Ai y? 0, the number system contains a modulus and

is either simple or semi-simple, that is, is constituted by v ̂ 1 mutually nilfactorial quadrates; and, conversely, in this case, Ai

= A2 ?? 0.

17 Comptes Rendus, 124, 1218 (1897).



652 proceedings of the American academy.

3.

It has been shown by C. S. Peirce that any given hyper complex number system (eh e2, ...

em) is a sub system of a quadrate of order n,

where the greatest value n need assume is ra + 1. This is, of course

equivalent to the theorem that any given number system can be

represented by a matrix whose order need not exceed m + l.18

Let now (ei, e2, ... em) be any given number system; let

euv (u, v = 1, 2, ... n) be the units of the quadrate of wdiich

(ei, e2, ... em) is a sub system, when we have

(61)

and let

Uv?vw ?

?UW) ?uvtv'w ? 0

(u, v, v, w = 1, 2, ... n; v ?? v) ;

(62) Z Z ojx)e? M=l V=l

(i =

1,2, ... m).

The units eh e2, ... em may then be regarded as represented, respec

tively, by the ra linearly independent matrices E\, E2, ... Em, where

Eif for i ? i,2, ... m, is defined by the system of equations,

(63) (fe', fe', ... fe') =

( 0iA 6?P, ... dlnU gx, fe, ... ?n),

fe(i), fe?, ft??

ft?!?, ft??, J1)

and any number x = Z ^'^ ?^ (ei> e*> e?0 by the matrix of the

?=i

linear substitution

(64) mm m

(?i', fe', fe') = ( Z <w?>, Z s***?, Z ^m(?) Kb fe, .. fe).

%=i i=l

Z 2*021?, Z 3?fta?, Z ̂ (*)

?=1 t=l i=l

Z *Al?, Z *A2(?), Z *An? t=l t=l i=l

18 Loc. cit., p. 221; also These Proceedings, 10, 392 (1875). In certain

cases, as shown by Peirce, we may take n<m\ in other cases, n must be

greater than m. See ? 4.




For any number of the quadrate em(u, v ? 1, 2, . . .

n) the two

scalar functions with respect to this number system defined in theorem I are equal as shown in ? 2 ; and, therefore, but a single symbol is re

quired for these functions. I shall denote by 5 A the two equal scalar functions of any number

(65) -4 ? 2- 2* Q-uv ?UV

?

W=l 0=1

#n, #12, . am

a2\, a22, ... 02?

#m, an2, ... an

of the quadrate; and, by theorem I, we then have

(66) Seuu = -, Seuv = 0 (u, v = 1, 2, ... n; v ?? u),19 n

and, therefore,

(67) SA Z Z auv S uv = - 21 au

U=l 0=1 w=l

I shall denote simply by 1 the modulus of the quadrate, and pi, for any scalar p, simply by p. We have

(68) l ? 2^ *uu"

_ n n

Any number A = 21 Z auv ew of the quadrate satisfies an equation U=l 0=1

(69) 4>U) =

{? -

Pl) (I -

pt) ... (? -

p?) =

0,

where the p's are scalars; and we have

(70) 4>[p) P ?

#11, ?

#12, ?

#m

- #21, P

- #22,

- #2tt

? #nl, #n2, P #nn

(p ?

Pl) (P ?

P2) (P-Pn)

19 For the number of linearly independent numbers X of the quadrate

satisfying the equation euuX = X is n, since every such number is linearly

expressible in eui, eW2, . . . Mn, and each of these numbers satisfies this equa tion. Therefore, by theorem I, n2Si Un = n. Similarly, the number of

linearly independent numbers X of the quadrate satisfying the equation Xeuu

= X is also n; and, therefore, n2/S2 uu = n. Since, for v ?? u, euv is nil

potent, Sl uv = S2 eUv = 0 (v t? u).

20 Cayley: Philosophical Transactions, p. 800 (1858).




The polynomial <?(p) is termed the "characteristic function" of A,

and (?)(p) = 0 the "characteristic equation" of A. Since, by (67),

n S A is the sum of the constituents in the principal diagonal of the

matrix representing A, it follows that n S A is equal to the sum of the

roots of the characteristic equation of A.

If A is idempotent, the roots of its characteristic equation are 0 and

1. Wherefore, if A is idempotent, n S A is equal to the mtdtiplicity of the root 1 of the characteristic equation of A.

In conformity with the notation employed in ? 2, let

(71) ? (2) ^ X?~ + pU^ + ... + p^? = o

be the syzygy of lowest order in powers of A. Then p<? (p) contains

S (p). Whence it follows that n is the maximum number of distinct non-zero roots of the equation ?2 (p)

= 0. Therefore, by theorem III,

and what was proved p. 636, A is nilpotent if, for some positive integer p,

~S Ap+h = 0 (h = 0, 1, 2, ... n -

1).

Conversely, by theorem I, if A is nilpotent, these equations are satisfied

for any positive integer p. n

For the scalar functions defined in ? 1 of any number A ? Z #? e\

?=i

of the system (e\, e2, ... em) I shall write Si A and S2A as in ? 1 and

? 2. The symbol S also is significant when prefixed to any letter de

noting a number of the system (e\, e2, ... em), since any such number

belongs to the quadrate euv (u, v} = 1, 2, ... n). We have, by (62)

and (67),

(72) Sei=1Z By?? (?= 1,2, ...n); 71 u=l

and, therefore m -j m n

(73) Sil = Z OiSei =-I I OiBvn?.

i=l n i=l u=l

Let now m m n n

(74) Z= Z x?ei= Z Z L *?0<?(i,e?.; ?=1 t = l W=l 0=1




and let the number system (e\, e2, ... em) contain at least one number

satisfying the system of equations

(75) SXei = xiSeiei+ x2Se2ei + ... + xmSemei = 0

(?= 1,2, ... m).

(76) V S^i^i, Se2ei, ... Semei

Se\e2, Se2e2, ... Seme2

Seiem, Se2em, Semem

we, therefore, now have V = 0, Let X = B be any number of

(ei, e2, ... em) satisfying equations (75). Then B is nilpotent; moreover, the product, in either order, of B and any number

m

A = Z akek of the system (eh e2, ... em) is also a solution of equa

k=l

tions (75). For, for any number

Y = yiei + y2e2 + ... + ymem

of the system (eh e2, ... em), we now have

SB Y = yi~SBei + y2SBe2 + ... + ymSBem

= 0:

wherefore, in particular,

SB*+h = ~SBBh+1 = 0 (h =

1, 2, ... n - 1),

and thus, by the theorem given on p. 654, B is nilpotent; further,

S(BA-ei) =

S(B'Ae{) =

0,

S (AB-ei) = S fa-AB)

= S(eiA-B)

= S(B-eiA)

= 0

(?= 1,2, ...m).

Since both B A and A B are solutions of equations (75), it follows

by what has just been proved that both B A and A B are nilpotent. In particular, for 1 = i ? m, Bei is nilpotent; and, therefore, by theorem I, SiBei

= 0. Whence it follows that B is a solution of the

system of equations

(77) SiXei =

XiSieiei + x2Sie2ei + . . . + .rmSi%^ = 0

(i =

1, 2, ... m).




Wherefore, we now have

Conversely, if B

Ai =

A2 =

0.21 m

Z o^e^ is any solution of equations (77),

Bcj (1 =j ~

m) is by theorem II then also a solution of these equa tions, and thus Bep by theorem I, is nilpotent: therefore, by the

theorem of p. 654, SBcj = 0 for j

= 1, 2, .. . m; that is, B is a solu tion of equations (75). Let the nullity of V be ml, where 1 = ml = m.

There is then a set of just m' linearly independent numbers

B\,B2, ... Bm> of the system (e\, e2, ... em) satisfying equations (75) ;

therefore, just ml linearly independent numbers of this system satis

fying equations (77) : whence it follows that the nullity of Ai is ml. x\nd since each of the jB's satisfies equations (77) it follows, from theorem II, that B\, B2, ... Bm> constitute an invariant nilpotent sub

system of (eh e2, ... em) containing every invariant nilpotent sub

system of (eh e2, . . . em).

Let now V ?= 0. In this case, if, for any two numbers

A a% e?? B = Z ha

of fe, e2, ... em), we have

SAet = SBei (i = 1, 2, ... m),

then A = B; otherwise, there is a number A ? B ^ 0 of the system

satisfying equations (77). In this case, Ai =^ 0 and the number

system (e\, e2, ... em) contains a nodulus but no invariant nilpotent

sub system.

Let now the number system (eh e2, ... em) be transformed by the

substitution

(78) e'i = 7iXei + n2e2 + ... + rimem

and let

(i= 1,2, ... m);

(79) v; S e'i e'j

(i, j =

1,2, ... m)

Then, since

_ mm _

S e'i e'j =

Z Z TihTjkSeh eu (i, j ?

1,2, ... m), h=l k=l

21 See p. 630.




we have

(80) V' = 2*V,

where T is the determinant of the transformation. Therefore, the

equation V = 0 is invariant to any transformation of the units of the

system (ei e2, ... em).

We have now the following theorem :

Theorem V. Let (eh e2, ... em) be any given number system consti

tuting a sub system of the quadrate euv (u, v = 1, 2, ... u) : thus let

ei = Z Z ?uv uv (i

= 1, 2, ... m). w=l v=l

For any given number

of the quadrate, let

n n

A ? 2^ L* ?uv?ui u=l v=l

SA = - 21 auu,

when, for any given number

m m n n

x = z xiei =

z Z Z xtoj^tu i=l i=l w=l v=l

of the system (e?f e2, ... em), we have

Let

SX=~Z 2; XiBn?. 71 ?=1 w=l

V = Saej

(i,j =

1,2, ... m)

denote the resultant of the system of equations

SXei =

? XjSejet = 0. (i

= 1,2, ...

m). y=i




Then, if the number system be transformed by the substitution

ei = Tji?i + ri2e2 + ... + Timem (i = 1, 2, ... m),

and if V

we have

Se'ie'j

(i, j =

1,2, ...

m)

V'= ^V,

where T is the determinant of the substitution. If V 5e 0, the system

(ei, e2, ... em) contains a modulus but no invariant nilpotent sub system;

and, in this case, if for any two numbers

?1 ? 2^ a\e%, B = Z bid

i=l

of the system we have

~SAei= SBc? (i =1,2, ... m),

then A = B. If V = 0 and m! (0 < m' tL m) is the nullity of V, the

system f (eh e2, ... em) contains a maximum invariant nilpotent sub

system with m! units constituted by any m! linearly independent solutions

of the equations SXei = 0 (i

= 1, 2, .. . m). In precisely the same way we may now prove the following theorem

of which the preceding theorem is a special case :

Theorem VI. Let (e\, e2, ... em) be any given hyper complex number

system constituting a sub system of the number system i, e2, ... en

whose constants of multiplication are yuvw for u, v, w ? 1, 2, . . . n, so

that

eu*v 2^ yuvw ?w ;

and let

Z Qiutu w=l

(i= 1,2, ...m).

For any number A = Z a" ? of the system (ei, e2, ...

en), let

w=l

SiA = - Z Z ?uTu Si A

u?\ v=l Z Z au7m

w=l v=l




in which case, for any number

m m n

A = ? aid =

][ ? aidiu u 1=1 ?=1 w=l

of the system (eh e2, ... em), we have

_ -. m n n _

Si A = - ? L Z ai?iu7uvv

71 i=l u=l ?=1

i m n n

?2^4 = " Z L L ?iftuTw

1=1 M=l ?=l

Finally, let X = J] a'???, awd Ze?

t=i

Vi = v2

= S2eiej

(i,j =

1,2, ... m)

i

(ij, = 1,2, ... m) \

be, respectively, the resultants of the systems of equations

(a) SiXd = xiSieiei + x2Sie2d + ... + xmSiemei = 0

(i =

1, 2, ... m),

(?) S2Xei = xiS2e2ei + x2S2e2ei + ... + xmS2emei = 0

(?= 1,2, ...m).

27??ft, ?f the number system (e\, e2, ... em) is transformed by the substi

tution e'i

= Tad + Ti2e2 + ... + nmem

(i =

1, 2, ... w), imd if

V'i ==

| Si e'i e'j

j (i, j = 1,2, ... m)

ice have

Vi' = ^Vi,

, V2

(i,j =

1,2, ... m)

Vr2= PV2

?r/ze/T l7^ ?/z<? determinant of the substitution. If Vi t^ 0, in tvhich case

V2 f^ 0, a??cZ conversely, the system (eh e2, ... em) contains a modulus,

but no invariant nilpotent sub system; and in this case, if for any two numbers

A = ? <*iei> B= I he-,

t=l




of this system, we have

'SiAei =

SiBet (i = 1, 2, . . . m),

or __ _

S2Aei =

S2Bet (i = 1, 2, . . . m),

then A = B. If the nullity of Vi is ml (0 < hi' = ?>i), in- which case

Vi = 0, the nullity of Vs ^ w'> ?^ conversely; and the system

(ei, e2, ... em) then contains a maximum invariant nil potent sub system

constituted by any m' linearly independent numbers of (eh e2, . . . em)

satisfying equations (a), or equations (?), every solution of equations (a) being a solution of equations (?), and conversely.

? 4.

Let (e\, e2, ... em) be any given number system; let eut!, for

u, v = 1, 2, ... n, constitute a quadrate of which (eh e2, . . .

em) is a

sub system; and let

n n

(81) ei= ? ? 6u?euv (i= 1,2, ... m).

The units of the system (eh e2, . . . em) are then represented, respec

tively, or may be identified, respectively, with the m linearly inde

pendent matrices defined by equations (63). The number system (e\, e'2, . . . e'm) reciprocal to (ei, e2. . . .

em)

is then also a sub system of the quadrate : that is,

n n

(82) e'i = ? ? W>eB, (i

= 1, 2, ... m) U=l V=l

for a proper choice of the rj's. For the m numbers e\, ef2, . . . elm of

the quadrate defined by equations (82) may be identified, respectively, with the m matrices E\, Efly . . .

E'm, where E\, for 1 ^ i = ?ra, is

defined by the equations

(83) E'i (?b fe, . - - ?n) =

( W?, W?), W? $?i, &, ... f?) ;

(i =

1,2, ... m);




and, therefore, if we put

(84) Wi} =

??.(i) (? =l,2,...m;u,v= 1, 2, ... n),

we then have

(85) E'i^tr.E?2 (i = 1,2, ...7/?):

whence it follows that E\, Ef2, . . . E'm are linearly independent, and

(86) E'iE'j =

tr.Ertr.Ej = fr. (?;?*)

mm m

= tr.(Yd 7jikEk)

= Z 7jiktr.Ek= Z 7iifc^'*

?=1 A=l ?=1

(?,j =

1,2, ... m);

that is to say, the numbers e\, ef2, . . . e'm of the quadrate are then

linearly independent, and

m

(87) e'i e'j =

21 7?*e'* fei = 1, 2, ... w).

?=1

We may take n = ???, and, at the same time, put

(88) Bw{i) =

7ivu, Vuv(i) =

7iuv (i, u, v = 1,2, ...

m),

unless, for a\, a2, . . . am not all zero, we have, simultaneously,

m m

(89) 21 O?7i?i = Z WO*? = 0, ?=1 1=1

(u, v = 1, 2, ...

m),

in which case, neither the m matrices E\, E2, ... Em of order m repre

senting, respectively, e\, e2, ... em nor the m matrices E\, E'2, . . . Efm

representing, respectively, e\, er2, ... e'm, are linearly independent.23

22 I here follow Cayley in denoting by tr. M the transverse (or conjugate) of

any given matrix M. Loe. cit., p. 31. 23 If n = m and $uv^

= 7ivu for i, u, v =

1, 2, ... m, the constituent of

EiEj in the nth. row and vXh column is m m m m

2 duw(i)9wv(j) =2 7^7, =2 7ijw7wm =

2 7ijw6uv(w) w=X w=l w=l w=l

by (54); and, therefore, m

^i^i =

2 7ijwEw.




In this case, there is some number .4 = Z cHei ̂ 0 of the system

*=i m

(ei, e2, ... em) such that A X = 0 for any number A' =

Z ^?<?? of t=i

this system; since we should then have

m m m

AX = Z Z Z O?Xjyijkek i=l y=i k=l

m m m

= Z Z (Z ciiyijk)xjeic= 0. j=l k=l i = l

m

Conversely, if A = Z ??^?5^0 and AX = 0 for every number A" of

?=i

(ei, e2, ... em), equations (89) are satisfied for at least one system of

values a\, 0%, ... am not all zero, and we cannot assign to the 0's, nor

to the 77's, the values given by equations (88). In this case, we have

SiAei = 0 =

S2Aei (i = 1, 2, ... m);

and, therefore,

Ax =

A2 = 0.

It is to be noted that equations (89) are the conditions, necessary and sufficient, that the reciprocal system shall contain a number

m m

A' = Z ??e'? y6- 0 such that X' A' ? 0 for any number X' =

Z ^% i=i

" t=i

of this system. Further, we may take n = m and put

(90) 0W? =

yuiv, r}uv(i) =7tiu (h u, v = 1, 2, ...

ra),

unless for b\, b2, ... bm not all zero, we have

m m

(91) Z &iT?. = Z biOu,? = 0 ?=i *=i

(w, r = 1, 2, ...

m),

in which case Eh E2, ... Em are not linearly independent, nor are



TABER.? SCALAR FUNCTIONS OF HYPER COMPLEX NUMBERS 663

E'i, Ef2, . . . Efm linearly independent.24 In this case, there is some m

number B = Z Mi ^ 0 of (eh e2, ... em) such that XB = 0 for

*=i m

every number X = Z x^i of this system; and there is also a number

?=i m

B' = Z M't ^ 0 of the reciprocal system such that Bf Xr* = 0

?=i

for every number X' of the reciprocal system. Conversely, if there is m

some number B = Z Mi ^ 0 of (<?i, ?2, ... em) such that XI? = 0

i=i m

for every number X of this system (or if, for Br = Z M'i ^ 0 and

?=i

for any number X7 of the reciprocal system, we have Bf X' = 0) equa

tions (91) are satisfied for some system of values b\, b2, . . ,bm not all

zero, and we cannot assign to the d's, nor to the 77's, the values given

by equations (90). When equations (91) are satisfied,

Srfei = 0 =

S2Bei (i= 1,2, ... m);

and, therefore,

Ai =

A2 = 0.

When the system (eh e2, ... em) contains a modulus it is not possible

to satisfy equations (89) nor equations (91). We may distinguish three cases. First, the given number system m

(e\, e2, ... em) may contain both a number A = Z aie% ̂ 0 and

?=i m

a number B = Z Mi ^ 0 such that AX = 0, XI? = 0 for every

1=1 m

number X = Z #?0? of the system, in which case the system does

?=1

not contain a modulus and Ai =

A2 = 0. In this case it is not possible

to assign to the d's the values given by either equations (88) or (90), nor to assign to the 77's the values given by either of these equations.

Nevertheless, it may be possible in this case to put n = m, provided m > 2, but not otherwise. Thus let m = 3, and let

ei2 =

eh eie2 = 0, eiez =

es,

24 If n = m and duv(i) =

7uiv for i, u, v = 1, 2, . . . m, it follows from (54)

thatE,j?Ty=2 7ijwEw. Cf. note 23. 10=1




e2c\ =

e2e2 =

e2ez =

0, ?3^1 = e3e2 =

^3^3 = 0:

if

A = a2e2 + a3es 5? 0, B = e2, we have

Aei= 0, CiB = 0 (i

= 1,2,3);

and w7e may now put n = m = 3, and

ei = en, c2 =

23, f3 = 13.

On the other hand, let m = 2 and let (c\> e2) contain a number A ^ 0 such that

A ei =

Ae2 = 0.

In this case, we may, without loss of generality, put A = e\, when we

have

Cl2 =

0, dC2 = 0.

If now ei =

Mi^, fl^A (i = 1, 2),

\021{i\ ?22(iV

we then have, since ei2 =

0,

(6nw, 0aW\ = n/0, fcXBT1,

W", 022(1V \0, 07 where & ̂ 0 and the determinant of the matrix cd is not zero; and, therefore, since e\e2

= 0,

/V2), 0?2>\ = us (a, j8\ ur\

Ui(2), fe(2v Vo, oy

where, without loss of generality, we may put a = 1, ?

= 0, giving

e2ci = e\, e22

= e2.

This system, however, contains no number B 9^ 0 for which

Cl B = e2 B = 0.

Second, the number system (??i, r2, . . em) may contain either a

number A 7^ 0 such that .1^- = 0 for i =

1, 2, . . . ra, or a number

B ?? 0 such that ^J5 = 0 for i = 1, 2, . . . m, but not both. In this

case, we may put n ? m and assign to the 0's and 77\s either the values

given by equations (90) or equations (88) respectively. Third, the system (ch e2, ...

em) may contain neither a number

A ?? 0 such that Aei = 0 for i =

1, 2, ... m nor a number B 9^ 0

such that ^2? = 0 for i =

1, 2, . . . m, for which a sufficient, but not




necessary condition, is the existence of a modulus, and, a fortiori,

that Ai 9a 0. In this case, we may put n = m and assign to the 0's

the values given by equations (88), and to the r?'s the values given by

equations (90). We then have

m m m m

A = Z ciiCi =

Z Z Z aiyivutuv, i?1 i=l w = l v = l

(92) m m m m

A' = Z aie'i = I Z ? ??7wm u?;

i=l ?=1 W=l ?>=1

and, therefore,

-j m m _

?>i^ = ? Z Z fliYiuM = *S^4,

(?3) S2^4 = ?

Z Z Oi7wu = ~SA'. m i=l u = l

On the other hand, if we assign to the 0's the values given by (90) and to the w's the values given by (88), which is now possible, we shall have

m m m m

A = Z ai?i =

Z Z Z wyuiv uv, . . ?=i ?=i w=i r=i

(94) m m m m

? = Z me'i = Z Z Z ?i7iu? w? ?=1 i = l u=l v=l

whence follows -, m m _

SiA = ? Z Z Oiliuu = S .4',

-, m m _

S2A = - Z Z Oi7tt?u = S 4.

1=1 w = l

When either the representation of the number system (ci e2, . . . em)

and its reciprocal system given by equations (88) or by equations (90) fails, and indeed in any case, we may proceed as follows. Let n =

m + 1, and let

(96 a) ew& = yivu, 6Wi/} =

ft?+W? = 0

(i, u,x>= 1,2, ... m),

(96 6) <Wi} = 0, W*'>= 1

(i, u ? 1,2, ... m; Ut??)\




moreover, let

(97 a) riuv? = Ywu, Wi/;

= Vm+l,m+lii}

= 0

(i, u, v = 1,2, ... m),

(976) Vum+i^ = 0 ^>+i^

= 1

(i, u = 1, 2, ... m; w ^ i).

The m matrices JEi, E2} ... EOT which we thus obtain have the same

multiplication table as the units of the system (ei, e2, . . . em) and

are, moreover, linearly independent. For, if

ciEi + c2E2 + ... + cmEm =

0,

then m

? aduv^ = 0 (u, v = 1, 2, ... m + 1);

t=i

and, therefore, in particular

m

Cu= L Ci0M,m+i(^ = 0 (u

= 1, 2, ... ra). i=i

Further, the m matrices determined by the above values of the t/s are also linearly independent and have the same multiplication table as the system (e\, er2, . . .

e'm) reciprocal to (eh e2, ... em). We now

have m m m m

A = Y. o,iet =

? ai(H Z 7ivuem+ eitm+i), t=l i=l u=l v=l

(98) m m m m

A' = 21 cuei =

Z a?(H Z IvinCuv + i>Bl^i); i=l i=l w=l ?=1

and, therefore,

-j m m i -,

Si.4 = J] ? aijiuu =-SA,

m i=l u=l m

-, m m _1_ 1 _

S2A = - ? H ai Juin = -S ^4'.

m < = 1 u?l m

We may also proceed as follows. Let n = m + 1, and let

(100 a) 6W& =

y un, du>m+fi =

em+i,m+i(i) = 0

(i, u, v = 1,2, ... hi),

(99)




(1006) ?W/> =0, dm+1/) = 1

(i, v = 1,2, ... m; v 5? i);

moreover, let

(101 a) 7]uv(i) =

yiuv, Vu,m+l{i) =

Vm+l,m+l{i) = 0

(i, u, v = 1, 2, ...

m),

(1016) 17?H-l,,(i) = ?> Vm+l/}= 1

(i, v = 1, 2, ... m; v 5* i).

The m matrices Ei, E2, ... Em thus obtained are linearly independent, as are also the m matrices E\% Er2, . . .

E'm; and the former have the

same multiplication table as the units of the system (e\, e2, ... em),

while the latter have the same multiplication table as the units of the reciprocal system. We now have

m m / m m

A = Z aiei = Z ai[ Z Z yuivtuv + TO+i.i

(102) m m / m m

A' = Z (He/= Z ai[JL Z 7iu? ut, + ?rn+i,?);

i=l i=l ^ u=l ?=1

and, therefore,

(103)

^ = ?Z Z 7?~^,

>V4 = - Z Z yuiu =->s^4.

The fundamental properties of the scalar functions given in theorem I

are more readily proved for the special case in which the number

system is a quadrate than in the general case. What precedes in this section indicates how the properties of these functions may be made to

depend upon the properties of the single scalar function of a quadrate.

Clark University,

Worcester, Mass.



On the Scalar Functions of Hyper Complex Numbers. Second Paper

Documents

Transcript of On the Scalar Functions of Hyper Complex Numbers. Second Paper