AND THEIR REIATIONSHIP }lith PLANE ROTATIONS

PROPERTIES OF REAL 2 X 2 ORTHOGONAL MATRICES AND THEIR REIATIONSHIP }lith PLANE ROTATIONS SAMJEL G. LiDLE* ad DAVID M. ALLEN* BU-546-Mt Jauary, 975 Abstract. We begi by partitioig all real 2 X 2 orthogoal atrices ito two fors: syetric ad asyetric. Certai k.~ properties of these atrices are stated. Properties regulatig sus ad products of these atrices are the developed. Syetric ad asyetric plae rotatios are the defied. Sice all real orthogoal atrices ca be represeted as a product of plae rotatios, all properties give previously for 2 X 2 orthogoal atrices are exteded to plae rotatios. Five tables of exaples are give.. Itroductio. Orthogoal atrices are fudaetal i the areas of liear odels ad experietal desig. They are highly used i liear algebra ad other related fields of pure ad applied atheatics. Products of orthogoal atrices are ofte used to reduce atrices to upper or lower triagular for or to bidiagoal or tridiagoal for. Products of eleetary reflectios ad plae rotatios are types of orthogoal atrices which have bee foud to be particularly useful i this regard. I this paper we ivestigate plae rotatios. Products of 2 X 2 orthogoal atrices are basic to this study. * Departet of Statistics, Uiversity of Ketucky, Lexigto, Ketucky 40506. This paper was writte while the authors were o leave at Corell Uiversity. t Paper No. BU-546-M i the Bioetrics Uit Mieo Series, Departet of Plat Breedig ad Bioetry, Corell Uiversity, Ithaca, New York 4853

- 2-2.!!!.!! Two z ~ Orthogoal Matrices. Oe ca easily verify that all real 2 X 2 orthogoal atrices ca be represeted i ei tber the sflllllletric orthogoal for s(e) = [ cos a si e si e ] = s(e + 2k) - or the asyetric orthogoal for A(e) = [ -si e si e] = A(e + 2k) where 0 s;; e < 2rT ad k is ay iteger. Further, easily verified properties iclude:. Property (i) The eigevalues of s(e) are ad -. (ii) The eigevalues of A(e) are cos 9 ± i si e. Property 2 (i)!s(e)! = - (ii) IA(e)l = Now we are ready to state ad prove properties o sus ad products of 2 X 2 orthogoal atrices. Property 3 L A.S(e.) is of the for s(e) ad Z ~ A(e ) i=l ~ ~ i=l is of the for A(e)

,.. 3 - Proof: L ~ s(e ) = = L x cos. e which is of the for s(e} iff ll ( ~ >..i ) 2 + ( ~ ~ si e ) 2 = i=l = ll but usig ai2 e + cos2 ei = ad cos(ej - ei) =j +si ej si e we have ~)..~ + LL)..i)..j cos(ej- e ) =. = i:fj ll The proof for t )..ia(9 ) is early idetical. i=l reduces to I the special case of Property 3 where e i = e for all i ll L i=l )..~ + LL )..i)..j cos(e3 - ei) = ifj or

- 4 - so that LAis(e) = * s(e) i=l L A A(e) = :: A( e) = which are the obvious results. Property 4 (i) TT A(ei) =A( Lei)= A(cp) i=l = where l: ei = cp + 2k "ad 0 s cp < 2rT i= (i) JT s(ei) = = S(a) = S( ) A(a) = A( ) if=2k+l if=2k. where a = l: (-l)-i ei = + 2k. ad 0 s < 2 i=l Proof: For 0 s: el, e2 < 2rT we have sie ] 2' 2 l s(e + e 2 ) = A(e + e ) = A(a ) cos{e + e ) 2 l 2

, - 5.. where e + e 2 a: + 2k ad 0 s; a: < 2. Cosequetly, repeatig we have JT A(t\).. A( L e ) = A(cp) = = where r. e = cp + 2k ad 0 s; cp < 2rr i=l Now si e2 J -2 si(e2 - cos(e 2 - ] = A(e - e ) 2 ad si(e + e2) l = s( e + e ):t -cos(e + e ) J 2 2

- 6 - So if is odd If is eve JT s(ei) = s(e)a(e 3 - e2)a(e 5 - e;:> A(e - e_) i=l = s(e - e + e )A(e - e 4 ) A(e - e ) 2 3 5 ~~ - = ' = s(e - e + e e ) = s(a) = s( ). 2 3 JT s(e) = A(e 2 - e)a(e 4 - e 3 )A(e 6 - e 5 ) A( e - e_) i=l = = A(a) =A( ) where a = ~ (-l)-ie ad a = + 2rrk where 0 ~ < 2rr. i=l i Corollary to Property 4 (i) s(e)s(e2) = A(e2 - e) (ii) s(e )A(e 2 ) = s(e + e 2 ) (iii) A(e )s(e 2) = s(e2 - e) (iv) A(e )A(e 2 ) = A(e + e2) Proof: Proved i proof of Property 4. I view of Property 4, oe could equivaletly call s(e)[a(e)] the odd [eve] product for of 2 X 2 orthogoal atrices. I this paper a p X p atrix R will be called a eleetary reflector (ER) if R ca be writte as

, - 7 - R =I- 2x(x'x)-x' (or I- 2ww') where x is a o-zero p x vector of real copoets (ad w = x/llxll). (See [], pages 39-44i [2], pages 230-244; [3], pages 48-50.) Also, for w = (si ~2, -cos ~/2) we have s(e) = I - 2ww' so s(e) is a eleetary reflector {Householder atrix) ad all 2 X 2 eleetary reflectors are of the for s(e). Cosequetly, the set of 2 X 2 eleetary reflectors is idetical to the set of 2 X 2 syetric orthogoal atrices. 3 Syetric ~ As~etric Plae Rotatios. A syetric plae rotatio is erely a ::>difica.tio of the idetity atrix by replacig certai eleets by the eleets of a syetric 2 X 2 orthogoal atrix (a eleetary reflector or Householder a.trix). Thus we will deote a syetric plae rotatio by r 8 (e) = I(ei,j) = j i si e s. j.... si e - r--, I J ~ " Siilarly, a asyetric plae rotatio is erely a ::>dificatio of the idetity atrix by replacig certai eleets by the eleets of a asyetric 2 X 2 orthogoal atrix. Cosequetly, we will deote a asyetric plae rotatio by

~' '" - 8 - J!'""'\ v. i.. si e IA (e) = I(ei,j) = A j.. -si e /"'"'.._J'. ' ' Equatios (43.) ad (43.2) o.page,47 of referece [3] is a type of eleetary uitary atrix which is a geeralizatio of syetric ad asyetric plae rotatios to iclude coplex eleets. Equatios (43.3) ad (43.4) o the sae page of this referece also defie a geeralizatio of asyetric plae rotatios to iclude coplex eleets. The ter plae rotatio will be used as a geeral ter to refer to a atrix of either the syetric or asyetric type. B.y covetio, the diesio will be ad i < j throughout. The superscripts i ad j deote the idex pair for which rotatio or rotatio ad reflectio occur. Whe abset, the diesio ad the idex pair (i,j) is iplied. The real arguet e refers to. the agle of rotatio whe we are cocered with asyetric atrices ad the agle of rotatio before or after reflectio i the case of syetric atrices. If several arguets appear this iplies a ew, possibly differet, agle of rotatio (or agle of rotatio before or after reflectio) for each idex pair provided all idices are distict. I particular let us exted our otatio to: Defiitio. Let B = (B,, ~)be a tuple of letters chose fro the set (s,a} ad let r, r 2,... r2k be distict itegers betwee ad iclusive,

, - 9 - the ' If all the copoets of! are of the sae letter type, we shall use the abbreviated covetio ot placig oly the letter type itself as subscript. Thus, for exaple, Plae rotatios ca be used to reduce a X p rectagular atrix to a upper triagular atrix usig approxiately 2p(p + ) ultiplicatios per row. This ethod is ost advatageous i the case of the availability of iial coputer storage sice space eed be allocated for oly oe row of the atrix at a tie. Properties l through 4 ca be odified to deal with these atrices as follows: Property ' (i) The eigevalues of I 8(e) are - oe's ad oe egative oe. (ii) The eigevalues of IA(e) are - 2 oe's ad + i si e ad :tts cojugate. Property 2' (i) II 8 (e)f = - ( ii) ria (e) I =

- 0 -., Property 3' E Xki8(ek) is of the for I 8(e) ad E \IA(ek) if of the for IA(e) iff k=l k=l ad ().E X = l k=l k Proof: It follows easily upo reviewig the proof of Property 3 ad the defiitios Property 3 If either \ > 0 for k -,, or \ < 0 for k =,, the.e ~ki 3 (ek) is of the for I 8(e) ad.e XkiA(ek) is of the for IA(e) iff k~. ~ ad (i I) r. A = k=l k (ii '.,) ek = e for all k. Proof: Fro Property 3', whe o coditios are iposed o the Ak's, E Aki8(ek) is of the for I8(e) ad.e >.kia(ek) is of the for IA(e) iff ~ ~ ad (i).e \.... k=l

Upo squarig both sides of (i), we have - ll-- Cosequetly, if (i) ad (ii) are to be cosistet, we ust have But uder the ore restrictive hypotheses of Property 3", we also have the coditio that either >..k > 0 tor all k or \ < 0 for all k. Sice the products \? t (k f t) are al'\-lays oegative uder this coditio, '\-le Illllst have that for all t r k. But sice o :s: ek, e t < 2rr, this iplies that all the ek 's are equal. Cosequetly, uder the restricted coditios of Property 3", () ad (ii) reduce to ad (') l: >.. = k=l k (ii') ek = e for all k.

... 2.. Property 4 ( ). I A ( e ) = I A ( L J..=l.IFl e ) = I A( co) where E e t = cp + 2k ad 0 s: cp < 2rr.= (ii) r I (e ) = il s.2.= I 8 (a) = I 8 ( ) IA (a) = IA ( ) f=2k+l f=2k where a= E (-l)-tet = + 2k ad o s:.< 2rr b=l Corollary to Property 4' (i) I 8(a )I8(e 2 ) = IA(e2 - e ) (ii) I 8 (e )IA(e 2 ) = I 8 (e + e 2 ) (iii) IA(e)r8(e2) = r8(e2 - e) {iv) IA(e )IA(e 2 ) = IA(e + e 2 ) It should also be oted that the product of two plae rotatios whose idices atch exactly oce will produce a atrix which is a odificatio of the idetity atrix by repla.cig certai eleets with the eleets of a 3 x 3 orthogoal atrix. For exa.q>le, for j < j'

- 3- i j j I 2 j.... -l si e si e 2 j I Si e 2 0-2. l Cotiuig i this way, a product of plae rotatios with the appropriate atchig idices ca produce ay orthogoal atrix. gj To ake the atrix :>re appealig to the eye, oly the o-zero eleets of this atrix are explicitly give i the array.

EXAMPlES - ~: I all of the followig exaples, S ca be replaced by IS ad A by IA so as to exted each fro 2 x 2 orthogoal atrices to plae rotatios. Replacig e with -e i either s(e) or A(e) has the effect ot chagig the off-diagoal sigs; replacig a by - e has the effect of chagig the diagoal sigs. Also S( + e) =.s(e) ad A( + e) = -A(e). I. SIMME'l.'IUC BY SIMMETRIC s(e)s(e) = A(o) = I s(e)s(.e) = A(-29) s{-e)s(e) = A(2e) s(e)s( - e) = A( - 29) s{ - e)s(e) = A(2e - ) s(-e)s( - e) = A() = -I s( - e)s{-e) = A(-) = -I r. I geeral, we have s(e)s(e2) = s(-e2)s(-e) = A(e2 - e) s(e2)s{e ) = A(e - e2). Thus, s{e)s(e2) F s(e2)s(e) ad ultiplicatio of 2 x 2 syetric orthogoal atrices by 2 X 2 syetric orthogoal atrices is ot coutative. II. ASYMMETRIC BY ASYMMETRIC A(S)A(e) = A(29) A(-e)A(-e) = A( - e)a( - e) = A(-29) A(S)A(-e) = A(-9)A{e) = A(O) = I A(S)A( - e) =.A( - S)A{.EJ'J : A() = -I A(-e)A( - e) = A( - S)A(-e) = A( - 29)

I geeral, we have - 5 - A(e)A(e2) A(e2)A(e) = A(e + e 2). Thus, ultiplicatio of 2 X 2 asyetric orthogoal atrices by 2 X 2 asyetric orthogoal atrices is coidi!iltati ve. III. SYMMETRIC BY ASlMMETRIC 8(e)A(e) = A(-e)8(e) = 8(2e) 8(-e)A(-e) = A(e)s(-e) = 8( - e)a( - e) = 8(-2e) A(e)8(e) = 8(9)A(-e) = 8(-e)A(e) = A(-e)8(-e) = A( - e)8( - e) = 8(0) = [; -~ 8(9)A( - e) = A{ - e)8(-e) = 8( - e)a(e) = A(-e)s( - e) = s(-) = s() s( -e )A( - e) = s( - e )A(-e) = A( e )s( - e) = s( - 2e) A( - 9)8(e) = 8(26 - ) = c-~ ~J I geeral, we have Thus, ultiplicatio of 2 X 2 orthogoal atrices, oe of which is syetric ad the other asyetric, is ot a coutative operatio.

- 6- IV. SPECIAL MATRICES Iet s... s( e) ad A = A( e) s = s3 = s5 = = s2k+l = 2 4 6 2k I= A(O) = S = S = 8 = = S = A'at = A(ate) = A(a) where 2lt9 = a + 2t ad 0 s a < 2rr s(-e)s(e) s(-e)s(e) 8(-e)s(e)... A(26)A(2e) A(2e) = A(2e) =A( ) where 2e = + 2lt ad 0 s <; 2rT. s(e) = [ co89 si e a a] - s(-e) = ( -si e -s a] - s(rr - e) =(- si e s e] = -s(-e) s( + e) =[- -sixi e -s e] = -s(e) s(rr/2 - e) [ si e J- -s e [ -si e s(rr/2 + e) = ] si e

s ( 3TT/ 2 - e ) = [ ~ -si e - - 7.;?.. = -s( TT/ 2 - e) si e... '~ ~-J~ - 9 l S(3rr/2 + e) = [ si e - - J- -si e = -S(TT/2 + 9) A( e) [ = -,~i e si e ] A(-e) = si e -si e ] A(TT - e) = [- -si e si e J- - = -A(-e) - A(rr + e) [.=_ si e -si e ].. = -A( e) - si e A(TT/2-9) [ = "fcos S. ] si e -si e A ( TT/ 2 + 9) = [.. - - ] -si e. A(3TT/2 - e) = l-si e - ] = -A(TT/2 - e) -si e

- 8 - A(3T/2 + e) [ si e - si e ] = -A ( / 2 + e ) V. OTHER EXAMPLES I4( 9 2,3)I4( 62,4) s s = 0 0 0 si e o -si e 0 0 0 0 0 0 0 0 o 0 si e 0 0 0 0 0 -si e 0 cos 9 0 0 cos 2 e o -si e 0 -si e 0 0 si e si e -si 2 e 0 REFERENCES (] H. R. SCHWARZ, H. RUTSHAUSER, ad E. STIEFEL. Nuerical Aal.ysis!?.! $yetric Matrices, traslated by P. Herteledy, Pretice-Hall, Eglewood Cliffs, New Jersey, 973 [2] G. W. STEWART. Itroductio ~ Matrix Coe-tatios, Acadeic Press, New York ad Lodo, 973 [3] J. H. WIIKSON. ~Algebraic Eigevalue Proble, Oxford Uiversity Press, Ely House, Lodo, W.l, 965.