Recreation Explorations in Determinants Roger Hui +.+(I.~)Pl tends to ( I ' ; ~ A . ~ ) - * I , because as wc s c c i n the following, going from one fine to an equivalent on the ilcxt, +.+(I,m)pl Ixl+~(~-I 1)Pl 1~1+(!~-1) x+/t!t@-I ~+(!i)~+/ 4.!t~-1 ((!~)+!~-1) +(!l)x+/i!tl-I (!~)~(+!1-1) ++/t!tu-1 (!m)~+/4.lt~ T h e dyad aU. ym derived from the dot operator is inner product. For example, a + . x~ is the matrix multipfication of finear algebra. A dictionary of A P L ([] 18 1) defines the monad O. Y~ on matrices ~ thus: O.V~: INDUCTIVE BZPOTRESIS x ! • DISTRIBUTES 4. O V E R +; : : (0{~1~) U . V $ l=-l{pao U/.I (<">">(0=ct0{pl)(~ x DISTRIBUTES ~; + ; + / OVER + T h e O-part of the direct definition, taken apart: (0(~i~) g. Y + / + ! t ~ is tim ~-telm lX)wer ~ n ~ expan~on of - 1 ( and IS (~Pl)+. -~ (~-1 "1"he fimL~t iS (--1--1+ ~nd~ m *I). R O W 0 O F q~; T H E F I R S T COI, rffNW OF DZADIC O. V (...) (~ SELF-REFERENCE OR RECURSIOW. ( , f A L L O W S UNN&~IED D I R E C T DEFINITIONS TO REFER TO THEMSELVES. ) A 3 -DIHENSIONAL ARHA~ OBTAINED B ~ E X C L U D I N G FRON II I N T U R N ROW 0 AND CO~CtMN O. ROW 1 AND COI, rffNlf 0 . ROW 2 AND CO[,DTIN O. AND SO ON. I.Jkewi~, + . - (~ , I ) P l 1)Pl, or ~ = 1 - ~ ( ~ - 1 1)P1. I)l ( ! ~)x*-l: +.-(m.m)pl uxl-S(m-1 1)Pl rex1- ( - 1 . m ) x ( ! w - l ) x- / 4. tm- t INDUCTIVE EYPOTEESIS This is expansion by minors along the flint column of ~. A model for particular functions O and V is straighffoward to implement in ISO A P L . - . x l is the ordinary determinant of linear algebra. Eugene McDonnell suggests the following simple but fruitful technique. If 0 and V are O: V: ' ('. (wa). ,-,. , (,. (~a). ,.,. (w~).,), (~,).,), w-rex ( - 1 . a , ) x ( , . , ~ - 1 ) x-/4. ! tm-1 ((:w) ,.w-1) - ( - 1 , m ) x ( :u )x-/+ : tm-1 (:u)x( !m-1) + (-1.w+I)x-/+ .' t m - 1 ( ! i )x (-1.w+1)x (-/4 ! tin-1 ) + ( : ~ - 1 ) x - l . u + 1 ( : ~)x (-1.w+1)x-/t : tM (In transforming the penultimate to the last, consider separately the cases m odd and w even.) - / v : l . w is the m-term power sexies expansion of * - 1 +.x(aJ.=)pl is (rap1)-i- l X~(U-1 1)Pl, or Then the ' O. Y~ computes a string of the symbolic formula for determinant. O. V 2 2P vABCD ' is ((A~D)-(C~B))', and U . V 3 3 P ' A B C D E F G E I ~ is ( (Ax ( (Exl)- (BxF)) )- ( (D- ( (Sxl)- (BxC)) )(Ox ( ( S x F ) - ( E x C ) ) ) ) ) u . ~ ( m - 1 1 ) p l . T h i s i s j u s t :m. +. t (I.m)Pl is (wpl)+. ~ (~-1 1 )P2, or Iv~(i-1 1 )Pl. A simple induction shows it is equivalent to 4./u-tin. Expressing this in "losed form" is an open problem; I c o n j e c m ~ o l ( P I ) is involved. +.+(m.u)pl +.-(m.m)pl +.x(u,w)pl + . 4-(m, u ) P l TENDS T0 Alternative definitions for O and V can exploit propertics such as associafivity or distfibufivity to simpfify the result. The technique can of c o u r ~ be applied in other areas. W e now look at the behaviour of four particular determ~nants, + . + ( ~ , ~ ) P l , +.-(~.~)Pl, +. + ( ~ . l ) p l , as functions of ~. +.x(I.~)pl, and TENDS TO ~-~ ~ (;re)M*1 (-l,l+W)x !u .t/~- tl ( !l)x*-I In closing, we offer the following mental cafisthenics for y o u r recreational con.Adoration: +.+(u.u)pl ~ +/x~u-tm (XEN IVERSON'S OBSERVATION ) F r o m the definition, + . + 1 l P 1 is just 1, and +.+(~,l)Pl is ( ~ P l ) + . + , f ( ~ - I 1 ) P l , or ~ x l + $ ( m - 1 1 ) P l . T h e first few values for this function are: +.+(l.I)pl 1 2 3 t~ 5 G 1 L~ 15 6q 325 1956 +.-(m,m)pl +.x(mjm)Pl +. i.(w,m)pl ~ ~ ~ -Ix\m-'Lm xl / m-'Lu m-tin x . EXPLORE O . V FOR F U N C T I O N S OTHER THAN + EXPLORE U . V ON ARRAYS OTHER T H A N (m.m)pl. A P L Q u o t e Q u a d 19 1 S e p t e m b e r 1988

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