Generalized Determinant

Transcription

1 Genelized Deteminnt Nikoly Dementev 1 Spotivny Steet, Olovk, Hohol Ae, Voonezh Region, 9686, Russi The epot suggests n ppoch to extend concept of deteminnt to the systems of ny ode. KEYWORDS: deteminnt, genelized, genelized deteminnt Intoduction Conventionlly, deteminnts e defined fo the sque mtices only. Sque mtices e gphicl epesenttions of the second ode systems, the systems tht depend on to pmetes. Fo exmple, second ode system cn be pesented s collection of ll its components nged in sque, the sque mtix: s s (1) ( ) In the cse the uppe index denotes the o nd the loe index denotes the column [1]. The mtix is x becuse in this pticul cse indices nge fom 1 to. Deteminnt of the thid ode (i.e. fo the x sque mtix) is defined s: s i, j, k 1 i j k (A) i, j, k i 1 j k (B) () hee i, j, k is pemuttion of 1,,. Plus sign is given to evey even pemuttion. Minus sign is given to evey odd pemuttion. Deteminnt cn be expnded in full eithe by columns (A) o by os (B) [1]. 1

2 In genel, second ode system cn be pesented s x sque mtix povided the indices nge fom 1 to. Deteminnt of such mtix ill be the deteminnt of -th ode: s 1,,, (1) () () () 1 1 (s1) s1, s, s, s (s) (s) (s) () hee 1,,,, o s1, s, s,, s is pemuttion of 1,,,,. Plus sign is given to evey even pemuttion. Minus sign is given to evey odd pemuttion. It should be stessed hee, tht even though thee exists n ppoch to define nd clculte the deteminnt of ny ode (see eq. () bove), the deteminnt of -th ode is conventionlly defined only fo the second ode systems. This epot suggests n ppoch to extend concept of deteminnt to the systems of ny ode.

3 d ode systems deteminnts In the spiit ith the conventionl definition of the tem deteminnt, hich s developed fo the nd ode systems, it seems esonble if one ould suggest folloing eqution s definition nd the clculting pocedue of deteminnt of the -th ode fo the d ode systems: s t 1,,, s1, s, s, s (1) () () () (s) (s1) (s) (s) 1,,, t1, t, t, t (1) () () () (t1) (t) (t) (t) (4) s1, s, s, s t1, t, t, t (s) (t1) (t) (t) (t) (s1) (s) (s) hee 1,,,, nd s1, s, s,, s o 1,,,, nd t1, t, t,, t o s1, s, s,, s nd t1, t, t,, t e the septe pemuttions of 1,,,, ithin ech of the pis of index sequences.

4 Fo instnce, -th ode deteminnt fo the d ode system cn be expnded in full by one of the index sequences (t-index sequence in this exmple) s: s t (1) () () (s1) (s) (s) = 1,, s1, s, s [ = ( )( 1 ) ( 1 ) 1 ( 1 ) ( 1 ) ( 1 ) 1 1 ( 1 ) ( 1 ) ( 1 ) 1 1 ( 1) ( 1) ( 1) 1 1 ( ( ( ( 1) ( 1) ( 1) ] (5) [ 1 1 ( 1 )( 1 ) ( 1 ) 1 1 ( 1 ) 1 ( 1 ) ( 1 ) 1 1 ( 1 ) ( 1 ) ( 1 ) ( 1) ( 1) ( 1) ( ( ( 1 1 ( 1) ( 1) ( 1) ] 4

5 n th ode systems deteminnts One could notice tht the indices of the components of the d ode systems (see eq.(4)) fom n indices mtix: 1 s1 t1 s t s ( ) t s t (6) Nottion of the components in the mtix cn be unified s: α1 α α α α1 ( ) 1α1 α 1α α 1α α 1α (7) hee evey numbe befoe α denotes pticul index sequence; evey numbe ight fte the α denotes pticul membe in the index sequence; α cts s septo. Thus, folloing the nottion of eq.(7), indices mtix fo the components of the system of n th ode should look like: ( nα1 nα nα nα α1 α α α α1 α α α 1α1 1α 1α 1α ) (8) 5

6 Thus, ccoding to the sequence of the guments, developed so f, the genelized deteminnt should be defined s: nα α α nα1, nα, nα,, nα 1 1α. α1, α, α,, α α1, α, α,, α (nα1) (α1) (α1) (nα) (α) (α) (nα) (α) (α) (nα) (α) (α) nα1, nα, nα,, nα. α1, α, α,, α 1α1, 1α, 1α,, 1α nα1, nα, nα,, nα. α1, α, α,, α 1α1, 1α, 1α,, 1α (nα1) (nα) (nα) (nα) (α1) (α) (α) (α) (1α1) (1α) (1α) (1α) (nα1) (nα) (nα) (nα) (α1) (α) (α) (α) (1α1) (1α) (1α) (1α) (9) [n-1]α1, [n-1]α, [n-1]α,, [n-1]α. α1, α, α,, α 1α1, 1α, 1α,, 1α ([n-1]α1) ([n-1]α) ([n-1]α) ([n-1]α) (α1) (α) (α) (α) (1α1) (1α) (1α) (1α) hee n denotes ode of the system; denotes ode of the deteminnt. 6

7 Concluding emks It seems esonble to nticipte futhe development of the ppoch descibed hee vi the ppliction of e-systems nd Konecke delts to eq.(9) in the sme fshion s it s peviously outlined fo the second ode systems [1]. Tht ould llo one, fo instnce, to deive ule of multipliction of to genelized deteminnts. Genelized deteminnts my lso cll fo evision of mth pptus in quntum mechnics. Indeed, it is ell knon, tht physicl quntities in quntum mechnics e epesented by sque mtices, hich e the nd ode systems []. This mens tht quntum mechnics in pesent fom is intinsiclly devised to del ith the nd ode systems only. Menhile, Specil nd Genel Theoies of Reltivity (STR&GTR) e intinsiclly devised to del ith the systems, hich odes e highe thn. One might hope tht extension of the mth of quntum mechnics to the highe ode systems ould eliminte the ode mismtch beteen quntum mechnics nd STR&GTR, thus mking the theoies mutully comptible, theefoe poviding the bsis fo thei futhe unifiction. Refeences 1. Applictions of tenso nlysis by McConnell, A. J., Dove Publictions, Inc., Ne Yok, SBN: Quntum Mechnics in Simple Mtix Fom by Thoms F. Jodn, Dove Publictions, Inc., (005), ISBN: