HW4 - MA/PH 607 Howell 9/14/2011 Homework Handout IV A. Let...

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MA/PH 607 Howell 9/14/2011 Homework Handout IV A. Let . U œ ! Ô × Ö Ù Ö Ù Õ Ø " # m# " " ' ' ' #3 m3 $! $! $! È È È È È È È È 1. Compute and . U U T 2. Verify that is unitary. U 3. Let be any orthonormal set of vectors (so ), and let U $ œ ß ß Ø l Ù œ e f e e e e e " # $ 4 5 45 e f b b b " # $ ß ß be given by c d c d b b b e e e U " # $ " # $ œ . a. Write out the complete set of formulas for the ’s in terms of the ’s. b e 4 5 b. Compute each and verify that is an orthonormal set (as Ø l Ù ß ß b b b b b 4 5 " # $ e f claimed in the lecture). c. Find the ’s in terms of the ’s. (Remember to make use of the fact that is e b U 5 4 unitary!) B. For each of the following matrices, find values for and so that the resulting matrix is + , orthogonal. (Multiple answers are possible.) ! " + , ß ß + , + , " " # # $ % È È C. Find a second and third row for A œ Ô × Õ Ø " # # $ $ $ ? ? ? ? ? ? so that the resulting matri is orthogonal, and then verify that your matrix satisfies B k k l r det and . A A A œ " œ T D. Find complex numbers and so that + , U œ + , $M%3 È È is unitary.
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Homework Handout IV page 2 E. Here we are considering traditional vectors in 3-dimensional space. Let T œ ß ß œ ß ß e f e f e e e i j k " # $ be the standard basis, and let U œ ß ß e f b b b " # $ be given by b i j b i j k b j k " # $ œ m # ß œ # M $ m œ % m $ and . 1. Compute the matrices , , and . M M M M TT TU UT UU 2. Find the matrix such that for every vector in space, and use it to A v A v v l Ù œ l Ù T U find when . l Ù œ $ m % m ' v v b b b T " # $ 3. How would you find the matrix such that for every vector in B v B v v l Ù œ l Ù U T space. 4.
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HW4 - MA/PH 607 Howell 9/14/2011 Homework Handout IV A. Let...

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