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formula sheet 2

# formula sheet 2 - Formula Sheet EIGENVALUES OF a b c d...

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Unformatted text preview: Formula Sheet EIGENVALUES OF a b c d . Setting p = a + d and q = det( A ) = ad- bc the eigenvalues are Î» 1 = p 2 + p p 2- 4 q 2 , Î» 2 = p 2- p p 2- 4 q 2 . The following diagram in the ( q,p ) plane summarizes the different eigenvalue regimes. For a copy of this diagram is available from the course web site on the syllabus page. ORTHOGONAL BASES OF FUNCTION SPACES 1. (a) Orthogonal basis of L 2 (- â€˜,â€˜ ] = { f defined on (- â€˜,â€˜ ]; R â€˜- â€˜ f 2 ( x ) dx< âˆž} , in sines and cosines of period 2 â€˜ and the constant function: n 1 , cos Ï€x â€˜ , sin Ï€x â€˜ ,......, cos nÏ€x â€˜ , sin nÏ€x â€˜ ,...... o , where, (with h f,g i = R â€˜- â€˜ f ( x ) g ( x ) dx ), h 1 , 1 i = 2 â€˜, D cos nÏ€x â€˜ , cos nÏ€x â€˜ E = â€˜, D sin nÏ€x â€˜ , sin nÏ€x â€˜ E = â€˜, n â‰¥ 1 . (b) Generalization to any interval [ a,b ] : 1 , cos 2 Ï€x b- a , sin 2 Ï€x b- a ,......, cos 2 nÏ€x b- a , sin 2 nÏ€x b- a ,...... , where, (with h f,g i = R b a f ( x ) g ( x ) dx ), h 1 , 1 i = b- a, cos 2 nÏ€x b-...
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formula sheet 2 - Formula Sheet EIGENVALUES OF a b c d...

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