Norm of the sum of complex vectors. Say that we have n complex-valued vectors z_i and we want to evaluate: norm(sum_(i=1)^n z_i)_2^2

tidones0r 2022-09-27 Answered
Say that we have n complex-valued vectors z i and we want to evaluate:
i = 1 n z i 2 2
Now, I know that for real vectors x i it holds:
i = 1 n x i 2 2 = i = 1 n x i 2 2 + i j x i x j
But when dealing with complex vectors, the last term with the inner product will be a complex number in general so I am not sure that this formula generalizes for this case. What am I missing?
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Answers (2)

espovilham7
Answered 2022-09-28 Author has 10 answers
For two complex numbers
z 1 + z 2 2 = ( z 1 + z 2 ) ( z 1 + z 2 ) = z 1 2 + z 2 2 + ( z 1 z 2 + z 1 z 2 ) = z 1 2 + z 2 2 + 2 R e ( z 1 z 2 )
easily extended to n.
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Zack Chase
Answered 2022-09-29 Author has 3 answers
Generalizing first answer to n terms, since z 2 2 = z z we have
i z i 2 2 = i z i j z j = i j z i z j = i z i z i + i j z i z j .
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