L1BKAKY 



NEW YORK 

 BOTANICAL 



GAHI>f:N 



ON DOUBLE POLYADICS, WITH APPLICATION TO THE 

 LINEAR MATRIX EQUATION. 



By Frank L. Hitchcock. 



DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY. 



TABLE OF CONTENTS. 



Page. 



1. Gibbs' Concept of dyadics and polyadics 355 



2. Polyadics as vectors in space of higher dimensions 357 



3. The fundamental identity of dot multiplication 358 



4. Double polyadics 360 



5. The Hamilton-Cayley equation for double polyadics 362 



6. The scalars as sums of cubic determinants 366 



7. Scalars in combination with the idemfactor 367 



8. Scalars as square determinants whose elements are polyadics . . 368 



9. An invariant property of the scalars 372 



10. Forms which show the polyadic character of the scalars .... 374 



11. Invariants regarded as products 378 



12. Star products 379 



13. Scalars formed by star multiplication 382 



14. Identities with the star idemfactor 383 



15. Illustrations of various types of multiplication 384 



16. Transformation from matrices to double dyadics 385 



17. Transformation back to matrices 386 



18. Character of the coefficients as algebraic polynomials 390 



19. The equation of extent unity and order two 391 



20. The equation of extent two and order two 393 



1. Gibbs' Concept of Dyadics and Polyadics. 



A system of Vector Algebra in A T dimensions following Gibbs is 

 based on a set of A 7 unit vectors d, e 2 , • • • , e# such that the dot products 

 e»- ejt are unity w 7 hen subscripts are equal, otherwise zero. Any other 

 vector a may be expressed in terms of the fundamental units, 



a = fljei + 02e 2 + • • • + cin^n- (1) 



Two vectors a and b written together with no dot or other sign between 

 them constitute a dyad ab called the indeterminate product of a into b. 

 A dyad e^eA; will be called a fundamental dyad or dyad unit. If we 

 have any dyad ab the first vector a is called the antecedent, the second 

 the consequent. Both a and b will be called factors of the dyad. A 

 dyad is zero when and only w r hen all products of scalar elements 

 ciibk are zero, that is, when one of the factors vanishes in all its ele- 

 ments. 



