irt igmet of MP- er to quetio - 7- Norml tre log { : MP Priipl tree: I MP II MP III MP Priipl iretio: { I { II { III Iitill uppoe tht i tre tte eribe i the referee tem ' i the me tre tte but eribe i other referee tem '''. If l i the trformtio mtri the ' l l. he trformtio mtri h to importt propertie: l l I etl ±. herefore et' I etl l l l etl Il etl et Ietl et I. hu ' he the me eigelue. ie the eigelue re lo the root of the polomil equtio I I I for them to be the me the polomil equtio I I I ot hge from oe referee tem to the other. It me tht I I I re irit. Let u tht α { { i umber. ee α i lo umber. hu α { { { { { { α. ie α α oe olue tht α. i mmetri the eigelue of mmetri mtrie re l rel. 5 ee olutio of quetio. If i the referee tem hol the e mut ho tht there eit ''' referee tem uh tht ' ' '. ie t let oe of thee orml tree mut be poitie t let oe of them mut be egtie. uppoe > <. li of ll ple prllel to from the ple here i efie up to the ple here i efie llo oe to olue tht the orml tree o thee ple r otiuoul from poitie lue to egtie lue. ie the ritio i otiuou oe of the ple prllel to ill iel ero orml tre. Let u ll thi ple β. herefore oe efie three mutull orthogol ple: oe of them i the ple here i efie; other oe i ple β; the thir ple mut he orml tre equl to ie the irit equtio mut hol.
If e oier ll ple perpeiulr to ple β ie the ple here i efie up to the ple here i efie the orml tree r gi otiuoul from egtie lue to poitie lue. ie the ritio i otiuou oe of the ple perpeiulr ple β iel ero orml tre. Let u ll thi ple γ. herefore oe efie gi three mutull orthogol ple: oe of them i the ple β; other oe i ple γ; the thir ple mut he ero orml tre ie the irit equtio mut hol. 7 Deformtio griet: u 8 ree tri teor: 7 I mll tri teor: t erti itt of time the iplemet fiel i oli i gie b u Determie: he ree tri teor. b he tri t poit. he orml tri log iretio { t poit. he gle hge betee the perpeiulr iretio { {. e he gle hge betee iretio { {. u u u
b {N { 8 85 8 { { N N 7 { { N { { 5 5 5 { { o 8 7. o { { { { { b b B b b { { { C he gle betee {B {C i ompute through {B {C BCo. hu o. he gle betee {b { i ompute through {b { bo. hu impl hek tht the irit re ifferet.
P... 5.55 7. 5.75 7. 5.55 5.75 5.75 5.75 57.85 C uppoe the tre teor i ilble the oite eigelue problem i I{ { here { {. herefore he otitutie reltio for ompletel iotropi mteril re: he tri teor multiplie b etor { be ritte he firt ro of { i gie b If equtio i ue the firt ro of { implifie to
5 imilrl the eo thir ro of { implif to hu proig tht the priipl tre iretio oiie ith the priipl tri iretio. 7 moulu: 5 pproimtio for >> 5 8 5 7. hu 757.8. Combiig the to reltio boe moulu:
irt eritie: Let B the B he firt eritie i ero he o o. I the rge o < < o the term B i ero ol if B he the me ig. Uull >. hu B. eo eritie: B B i If > the B >. > > ii or B > B > < B B B Coit i egtie. ee it i mimum. mple: P 5 P.5 P > P.
7 i If B < the B <. < < B ii or B < B < > B B B mple: P 5 P.5.5 P <. P.
i If < B > the B. I thi e there re o etreme poit i the rge o < < o. < B > < > mple: P 5 P.5 P. he < P >. P 8
Mot tpil ompoite he miimum lue for i the rge o < < o. I the e of uiiretiol lmie the miimum lue i ot igifitl ifferet from. γ γ k 5 5 5 5 5 γ k