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# HW1 - A T B with tr C = n X i =1 c ii c = c ii 1 ≤ i ≤...

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Electrical Engineering and Computer Science Department University of Central Florida COT 6600– Quantum Computing Fall 2010 (dcm) Homework Assignment 1. – Due Wednesday, September 22, 2008. Problem 1 (20 points) Show that: e = cos θ + i sin θ and ( e ) n = cos( ) + i sin( ) . Hint: Review Taylor series expansion of a function before attempting to prove Euler’s and De Moivre’s formulae. Problem 2 (20 points) Show that C m × n the set of all m × n matrices with complex entries is a complex vector space. Hint: Review the properties of a complex vector space and show that V, W C m × n each one of the properties is satisfied. Problem 3 (20 points) Find the transpose, W T , the conjugate, W * , and adjoint, W of the complex matrix W : W = 6 - 3 i 2 + 12 i - 19 i 0 5 + 2 . 1 i 17 1 2 + 5 i 3 - 4 . 5 i Problem 4 (20 points) Given matrices A, B, C R n × n (square matrices with real elements) the inner product h A, B i is defined as:

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Unformatted text preview: ( A T B ) with tr( C ) = n X i =1 c ii , c = [ c ii ] 1 ≤ i ≤ n. Construct the inner product of A = 6 2 4 13 9 11 5 17 1 3 8 10 17 16 15 14 and B = -6 2-4 13 9-11 5-17-1 3-8 10 17-16 15-14 Given matrices V,W ∈ C n × n (square matrices with complex elements) the inner product h V,W i is deﬁned as: h V,W i = tr ( V † W ) . Construct the inner product of V = 6 + i 1 + 2 i 3 + 4 i 3-i 9 1-2 i 2 + 3 i 1-i i 3-2 i 8 5-4 i 1 + i 7 6 + 5 i 9-7 i and W = -6 + i-i 2-i 4 + i i-1-i-1 2 i 8-7 i 10-i 2 + 5 i-7 7 i Problem 5 (20 points) Construct the tensor and the outer products of the matrices A and B in Problem 4....
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