Emil Olteanu-The plane rotation operator as a matrix function THE PLANE ROTATION OPERATOR AS A MATRIX FUNCTION. by Emil Olteanu

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1 Emil Oltu-Th pl rottio oprtor s mtri fuctio THE PLNE ROTTON OPERTOR S MTRX UNTON b Emil Oltu bstrct ormlism i mthmtics c offr m simplifictios, but it is istrumt which should b crfull trtd s it c sil crt cofusios ormlism is istrumt which, togthr with progrmmig lgug tht llows bstrctios (for istc th lgug) c crt vr strog progrmmig istrumt O mpl is rlizig th mtri fuctios i lgug usig poitrs Th prst mtril rfrs to mpl of mtri fuctios, th potil mtri fuctio, usd s pl rottio oprtor mtri fuctio is fuctio of th form: Y ( X ), whr X d Y r qudrtic mtrics of ordr Th rltio c lso b writt s follows: th cs of th pl rottios w shll us mtrics of ordr d for rottios i thr-dimsio spc w us mtrics of ordr 3 ordr to ssocit mtri with lir trsformtio of -dimsiol vctoril spc V, w must choos bsis,,, rom th qutios: or ( j ) i ij i ( ( L ( ) ) ), for j,,,

2 Emil Oltu-Th pl rottio oprtor s mtri fuctio w obti th mtri which rprsts th trsformtio : f is th trsformtio obtid through th rottio of th pl roud th origi O with gl ϕ d if d r two vctors from th bsis orthogol d of lgth, th th rprsttivs of ths vctors pplid i th origi ppl to th rprsttivs of imgs ( ) d ( ) ( ) ( ) ϕ ϕ O igur ( ) d ( ) vrif th followig qutios: ( ) ( ϕ ) + si( ϕ ) ( ) si( ϕ ) + ( ϕ ) Th oprtor is rprstd b th mtri : 34

3 Emil Oltu-Th pl rottio oprtor s mtri fuctio si ( ϕ ) ( ϕ ) ( ϕ ) ( ϕ ) si Th oprtor - is th rottio of glϕ i th opposit dirctio, tht is of gl ϕ si ( ϕ ) si( ϕ ) ( ϕ ) ( ϕ ) si ( ϕ ) si( ϕ ) ( ϕ ) ( ϕ ) w vrifid th rltio U, th uit vctor si ( ϕ ) si( ϕ ) ( ϕ ) ( ϕ ) si ( ϕ ) si( ϕ ) ( ϕ ) ( ϕ ) si ( ϕ ) + si ( ϕ ) ( ϕ ) si( ϕ ) si( ϕ ) ( ϕ ) ( ϕ ) ( ϕ ) ( ϕ ) si( ϕ ) si ( ϕ ) + ( ϕ ) U Y Giv vctor X i pl w c obti vctor Y rottd Y b gl towrds which w ppl th pl rottio oprtor Y ( ) X, whr: ( ) si ( ) si( ) ( ) ( ) Th product of two gl rottios d β is quivlt to gl rottio + β This is mthmticll prssd s follows: ( ) ( β ) ( + β ) Th fuctio tht vrifis this fuctiol qutio is th potil fuctio This lds us to th coclusio tht th rottio oprtor c b prssd s potil ordr to fid th potil fuctio corrspodig to this oprtor w srch for diffrtil qutio to vrif d th w look for its solutios ( ) si ( ) si( ) ( ) ( ) 343

4 Emil Oltu-Th pl rottio oprtor s mtri fuctio ( ) si' ( ) ( ) ' ( ) ( ) ( ) ( ) si( ) ( ) si( ) ( ) ( ) ' si ( ) ' si' si whr w otd W obtid th qutio: '( ) ( ) writt ( ) ( ) B This qutio c lso b ' Grll spkig B, but i this cs th multiplictio is commuttiv rom th qutio: ' ( ) ( ) ( ) ' ( ) ( ) ( ) ' ( ) d d l ( ) d( ) d ( ( )) + + ( ) for ( ) Th formul c lso b writt: ( ) ( ) ( ) ( ) whr ( ) U W mk ssumptio ( ) Rsults: 344

5 Emil Oltu-Th pl rottio oprtor s mtri fuctio l W clcult l usig th dvlopmt i sris of potils ( ) + l U, whr w otd with U, th uit mtri d with mtri vribl 4 3 or O, whr O is zro mtri, rsults: ( ) + l O U Thrfor O, mig i, for i {,, 3, 4} Thus: ( ) ( ) ( ) ( ) ( ) si si This formul c lso b writt i othr mr: ( ) ( ) U + si, which is othr w of writig Eulr s formul i th fild of mtri fuctios W otic tht th followig rltio is tru: U

6 Emil Oltu-Th pl rottio oprtor s mtri fuctio Rfrcs [] W cs - omplmt d mtmtici cu plicţii î thică (Mthmtics omplmts with Tchicl pplictios), Editur Thică, Bucurşti, 98 [] Gh Şbc - Mtmtici spcil (Spcil Mthmtics), Editur Didctică şi Pdgogică, Bucurşti, 98 [3] rstici Borislv (coordotor), ş - Mtmtici spcil (Spcil Mthmtics), Editur Didctică şi Pdgogică, Bucurşti, 98 [4] Ghorgh om ş - troducr î tori cuţiilor oprtoril (troductio to th Thor of Oprtor Equtios), Editur Dci, luj-npoc, 976 [5] M Glfd şi G E Şilov - ucţii grlizt (Grlizd uctios), Editur Ştiiţifică şi Eciclopdică, Bucurşti, 983 [6] zlollh Rz - Spţii liir (Lir Spcs), Editur Didctică şi Pdgogică, Bucurşti, 973 uthor: Emil Oltu, Dcmbri 98 Uivrsit of lb uli, Romi, oltu@ubro 346

Linear Algebra Existence of the determinant. Expansion according to a row.

Linear Algebra Existence of the determinant. Expansion according to a row. Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit