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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 λ 1 = p 2 + p p 2- 4 q 2 , λ 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 (- ‘,‘ ] = { f defined on (- ‘,‘ ]; R ‘- ‘ f 2 ( x ) dx< ∞} , in sines and cosines of period 2 ‘ and the constant function: n 1 , cos πx ‘ , sin πx ‘ ,......, cos nπx ‘ , sin nπx ‘ ,...... o , where, (with h f,g i = R ‘- ‘ f ( x ) g ( x ) dx ), h 1 , 1 i = 2 ‘, D cos nπx ‘ , cos nπx ‘ E = ‘, D sin nπx ‘ , sin nπx ‘ E = ‘, n ≥ 1 . (b) Generalization to any interval [ a,b ] : 1 , cos 2 πx b- a , sin 2 πx b- a ,......, cos 2 nπx b- a , sin 2 nπx b- a ,...... , where, (with h f,g i = R b a f ( x ) g ( x ) dx ), h 1 , 1 i = b- a, cos 2 nπx b-...
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