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Math_235-Assig_9-F09

Math_235-Assig_9-F09 - Math 235 Assignment 9 Due Wednesday...

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Unformatted text preview: Math 235 Assignment 9 Due: Wednesday, Dec 2nd 1. Let 1? 2 (—2 — 32', 2 +2) and 172 (4 — i, 4 + Use the standard inner product on C" to calculate < 71,172 < 17,7] >, and 2. Determine which of the following matrices is unitary. a)A_[(1+¢)/ﬁ ~5/x/ﬁ] ‘ (1+2i)/\/7 (3+i)/\/ﬁ’ _ (Him/6 (1+i)/ 3 b)A‘i 2¢/¢6 i/\/§ ' 3. a) Verify that 12‘ 2 (1 + i, 1, 2) is orthogonal to 172 (1 — i, 22', 0). b) Determine the projection of U7 2 (2 + 2', 2 ~— 2‘, 1 — 22') onto the subspace of C3 spanned by 11' and 17. 4. Let V be a complex inner product space. Prove that “117+ g + 5. Prove that for any n x n matrix, detA 2 det A. 6. Prove that if U is unitary, then |det U} 2 1. Give an example Where det U is not equal to 1 or -1. 7. Let A* 2 (AF be the conjugate transpose of A. Prove that (A*)* 2 A, (aA)* 2 ‘dA* and (AB)* : B*A*. Use MATLAB to complete the following questions. You do not need to submit a printout of your work. Simply use MATLAB to solve the problems, and submit written answers to the questions along with the rest of your assignment. Complex Inner Products and Unitary Matrices Question 1 Note that the dot command in MATLAB calculates the standard inner product in C” when the arguments are complex. Leta: (2—i,3,i,1+z‘,—4+i) andb: (—2i,3—5i,—4,1—i,—2—3i). (a) Calculate (a, b). (b) Calculate (b,a). (C) Is this inner product symmetric? Question 2 Note that the transpose operator (') in MATLAB calculates the conjugate transpose when the matrices have complex entries. 1—1’ 1' 0 ~1+3i2+2i —2 75>" 7%? LetA= 2+2i 22' —2i andletB: *gﬁgﬁg 2 22‘ 1+4z' H. 2 41. To 7T6 m (a) Is A unitary? (b) Is B unitary? ...
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