# Inner product of complex vectors

by weetabixharry
Tags: complex, product, vectors
 P: 108 I have three (N x 1) complex vectors, a, b and c. I know the following conditions: (1) a and b are orthonormal (but length of c is unknown) (2) c lies in the same 2D plane as a and b (3) aHc = x (purely real, known) (4) bHc = iy (purely imaginary, unknown) where (.)H denotes Hermitian (conjugate) transpose, i is the imaginary unit and x,y are real numbers. Given that I know x, can I deduce y? My hunch is that (without the "purely real/imaginary" statements), these conditions would define y up to an arbitrary complex phase, but the "purely real/imaginary" conditions allow the phase to be known too. However, my reasoning relies on there being some sense of "angle" between a and c and between b and c... such that these angles sum to 90° for the orthonormality condition (1). I don't know if this is valid.
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P: 26,148
hi weetabixharry!
 Quote by weetabixharry (2) c lies in the same 2D plane as a and b
doesn't that mean that c must be a linear combination of a and b ?
P: 108
 Quote by tiny-tim doesn't that mean that c must be a linear combination of a and b ?
Yes, where I guess the coefficients of the linear combination are complex scalars.

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P: 9,839
Inner product of complex vectors

 Quote by weetabixharry Yes, where I guess the coefficients of the linear combination are complex scalars.
From which it follows that y=0?
P: 108
 Quote by haruspex From which it follows that y=0?
Why?
P: 108
 Quote by haruspex From which it follows that y=0?
c is a linear combination of a and b:

c = Aa + Bb

for A,B complex scalars.

Therefore, from (3) and (1), A = x
and, from (4) and (1), B = iy

I can't see why y=0

I guess, from this I have:

c = xa + iyb

which is 1 equation in 2 unknowns (y and c)... so I'm stumped.
 Homework Sci Advisor HW Helper Thanks P: 9,839 You know facts about aHc, bHc. How can you combine that with with knowing c = Aa + Bb? Actually I was wrong to suggest y=0, but you can at least make progress this way.
P: 108
 Quote by haruspex You know facts about aHc, bHc. How can you combine that with with knowing c = Aa + Bb? Actually I was wrong to suggest y=0, but you can at least make progress this way.
I combined these in my previous post to write A,B as functions of x,y.
Beyond that, I guess I can say:

|c|2 = x2 - y2
 Homework Sci Advisor HW Helper Thanks P: 9,839 Suppose you found a y and c =xa+iyb which satisfied all the conditions. Wouldn't 2y and c' =xa+i2yb also satisfy them?
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P: 26,148
hi weetabixharry!

(just got up )
 Quote by weetabixharry c = xa + iyb
that's right!
 which is 1 equation in 2 unknowns (y and c)...
so what's the answer to … ?
 Quote by weetabixharry Given that I know x, can I deduce y?

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