generalized conjugate matrices

Annals of Mathematics

Generalized Conjugate MatricesAuthor(s): Philip FranklinSource: Annals of Mathematics, Second Series, Vol. 23, No. 1 (Sep., 1921), pp. 97-100Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1967790 .Accessed: 24/05/2014 00:44

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].

.

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals ofMathematics.

http://www.jstor.org

This content downloaded from 193.104.110.32 on Sat, 24 May 2014 00:44:51 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=annalshttp://www.jstor.org/stable/1967790?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp

GENERALIZED CONJUGATE MATRICES. BY PHILIP FRANKLIN.

The notion of conjugate matrices, which originated with 0. Taber* and was applied in a recent paper by A. A. Bennett, t may be described in the following terms. If, corresponding to a given matrix M1 of order n there exist n - 1 matrices of the same order satisfying the conditions:

1. They have the same characteristic equation as Ml; 2. They are commutative with respect to multiplication; 3. The symmetric functions 2;Mi, 2MiMjM, * M1lM2 A Mn

formed from the matrix M1 and these n - 1 matrices are scalars and equal to the corresponding functions of the n scalar roots of the characteristic equation of M1;

these n - 1 matrices are called the n - 1 conjugates of M1. In the case where the roots of the characteristic equation of the given

matrix are all distinct, the existence of the conjugate matrices is demon- stratedl by noting that if ri, r2, * . r. are these n distinct roots, the matrix

ri 0 ... 0 o r2 *-- 0

(1) Rw

has the same elementary divisors as the given matrix M1. Consequently? a non-singular matrix P can be found such that: (2) M1 = PRP-1, and the n - 1 matrices (3) Mi = PRiP-11 where Ri is given by:

ri 0 0 0 0 I 0 ri+ .. 0 0 . . 0

(4) Ri=1O 0 ... rn 0 0. , 0 0 0.. 0 ri ... 0

10 0 .. 0 0 evidently satisfy the three conditions stated above.

* Taber, O., On certain identities in the theory of matrices. Amer. Journ. Math., vol. 13 (1891), p. 159.

t Bennett, A. A., Some algebraic analogies in matric theory, these Annals, vol. 23, p. 91. 1 Cf. Taber, 1. c. ? Bocher. M., Introduction to Higher Algebra, p. 283.

97


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

98 PHILIP FRANKLIN.

The same method applies to a matrix whose characteristic equation has equal roots, provided it has the same elementary divisors as a matrix of the form R1, with some of the r's equal. In the case of matrices whose characteristic equations have equal roots, but whose elementary divisors are not of this type, the method fails. In fact in this case there may fail to exist a set of n - 1 matrices conjugate to the given matrix.*

The purpose of this note is to define a set of generalized conjugate matrices, which are subject to less stringent conditions than the ordinary conjugate matrices, but which have the advantage of existing in all cases. Furthermore, they are sufficient for many of the proofs given by Taber and Bennett which use ordinary conjugate matrices.

Our generalized conjugate matrices differ from the ordinary conjugate matrices in that they are required to satisfy conditions (2) and (3) only. The sacrifice of condition (1) is not as violent as it at first appears, since by the use of the remaining two conditions the characteristic equation:

(5) Ml-xI = 0 reduces to: (6) (M1-)(M2- .) (Mn- X) = 0

and consequently the generalized conjugate matrices are roots of the characteristic equation.

To set up these matrices explicitly, we shall first noticet that any matrix is equivalent to a matrix of the form:

S2~

(7)

where the missing elements are zeros and the S1's are blocks of terms given by:

ri 1 0 0.. 0 0 ri 1 *.. 0

(8)

0 0 0 ... ri Simple roots correspond to blocks of a single term. We therefore have, analogously to (2): (9) ml = PS1PTl.

* Bennett, 1. c. t Bocher, 1. c., p. 289.



GENERALIZED CONJUGATE MATRICES. 99

In the case where Si consists of a single block,

rl 1 0 .. I o ri 1 ... 0

(10)

0 0 0 .. r the n - 1 matrices: (11) M= PSiP'l, where Si is given by

~r1 0 0 0 r1 o ~i- i- 0

(12) Si-

0 0 0 ... ri

w being a primitive nth root of unity, are evidently a set of n -1 ordinary conjugates of M1. In the general case, where M1 satisfies the relations (9) and (7), we form the n - 1 matrices:

S.2

(13) Si

where for each block Sii of k rows, the k - 1 conjugates of Sli, the corresponding block of Si, appear in k - 1 of the matrices, and in the remaining n - k blocks its place is filled by a scalar block of the form:

ri(j) 0 ... 0 0 ..0

(14)

0 0 *.. ri(* )

the ri's being roots of the characteristic equation of M1 and so selected that the n - k values of ri for the n - k different values of i together with the k roots corresponding to Sli form the complete set of n roots of the characteristic equation of M1. The generalized conjugate matrices of M1 are then the n - 1 matrices obtained by combining (13) and (11). This is readily verified if we first notice that any rational integral function of the Si's, f(Si, S2, * . Sn), is given by:



100 PHILIP FRANKLIN.

(15) f(S1, S2, . . Sn) =

f(Sll, ~ ~ f(~2 822, *. S

.S )) I f(S121 S922 . . . Sf 2)

As an illustration, the matrix: 10

010 0 02

has no matrices conjugate to it in the ordinary sense, but has as its generalized conjugates:

1 -1 0 I 2 0 0 0 1 0 and 0 2 0. 0 0 1 00 1

It is to be noticed that the method of constructing the generalized conjugate matrices leads to a unique result only in exceptional cases, like that just given. For example, in addition to the ordinary conjugates, the matrix:

0 b O0

has as generalized conjugates any of the pairs:

'6 0 0 C 0 0

0 aO b O O

0 c , 0 a O; O 0b O 0a

or:

0 a 0, 0 c O. OICTa 0N06S

PRINCETON UNIVERSITY.



Article Contentsp. 97p. 98p. 99p. 100

Issue Table of ContentsAnnals of Mathematics, Second Series, Vol. 23, No. 1 (Sep., 1921), pp. iii-iv+1-100Front Matter [pp. ]Volume Information [pp. ]Errata [pp. iv]On Matrices Whose Elements Are Integers [pp. 1-15]An Algorism for Differential Invariant Theory [pp. 16-28]The General Theory of Cyclic-Harmonic Curves [pp. 29-39]More Theorems on the Complete Quadrilateral [pp. 40-44]A Theorem on Cross-Ratios in the Geometry of Inversion [pp. 45-51]The Condition for an Isothermal Family on a Surface [pp. 52-55]The Reversion of Class Number Relations and the Total Representation of Integers as Sums of Squares or Triangular Numbers [pp. 56-67]Note on the Term Maximal Subgroup [pp. 68-69]Reducible Cubic Forms Expressible Rationally as Determinants [pp. 70-74]Note on the Picard Method of Successive Approximations [pp. 75-77]A Fundamental System of Covariants of the Tenary Cubic Form [pp. 78-82]The Modular Theory of Polyadic Numbers [pp. 83-90]Some Algebraic Analogies in Matric Theory [pp. 91-96]Generalized Conjugate Matrices [pp. 97-100]

generalized conjugate matrices

Documents