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 1 = p 2 + p p 2- 4 q 2 , 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 x , sin x ,......, cos nx , sin nx ,...... o , where, (with h f,g i = R - f ( x ) g ( x ) dx ), h 1 , 1 i = 2 , D cos nx , cos nx E = , D sin nx , sin nx E = , n 1 . (b) Generalization to any interval [ a,b ] : 1 , cos 2 x b- a , sin 2 x b- a ,......, cos 2 nx b- a , sin 2 nx b- a ,...... , where, (with h f,g i = R b a f ( x ) g ( x ) dx ), h 1 , 1 i = b- a, cos 2 nx b-...
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formula sheet 2 - Formula Sheet EIGENVALUES OF a b c d ....

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