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hw5 - Math 260 Spring 2012 Jerry L Kazdan Problem Set 5 Due...

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Math 260, Spring 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Feb. 16. Late papers will be accepted until 1:00 PM Friday . Unless otherwise stated use the standard Euclidean norm . 1. a) Let f ( x ) = braceleftBigg 1 for π x < 0 , 1 for 0 x < π . Find its Fourier series (either using trig functions or the complex exponential). b) There is an extension of the Pythagorean Theorem that expresses bardbl f bardbl 2 in terms of the Fourier coefficients. What does that assert for this function? c) Let g ( x ) = braceleftBigg 0 for π x < 0 , 1 for 0 x < π . Find its Fourier series. 2. For complex vectors Z = ( z 1 , z 2 , . . . , z n ) recall that we used the inner product ( Z, W ) := z 1 ¯ w 1 + · · · z n ¯ w n . Say you have an n × n matrix A := ( a jk ) whose elements a jk may be complex numbers. Define its adjoint A by the rule ( A Z, W ) = ( Z, AW ) for all complex vectors Z , W . Find a formula for the elements of A in terms of the elements of A . [First do the 1 × 1 and 2 × 2 casesz.] 3. A square real matrix is called symmetric (or self-adjoint ) if A = A . It is called anti-symmetric (or skew-adjoint ) if A = A . a) If a real matrix is anti-symmetric, show that the elements on its diagonal must all be zero. b) If A is a square real matrix, shot there is a unique symmetric matrix A + and a unique anti-symmetric matrix A such that A = A + + A . [Find formulas for A + and A in terms of A and A .] c) Find a symmetric 3 × 3 matrix A so that ( X, AX ) = 3 x 2 1 + 4 x 1 x 2 x 2 2 x 2 x 3 .
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