Inner product of complex vectors

In summary, the conversation discusses three complex vectors, a, b, and c, and four conditions regarding their relationships. The conditions state that a and b are orthonormal, c lies in the same 2D plane as a and b, aHc is purely real and known, and bHc is purely imaginary and unknown. The conversation also explores the possibility of deducing the value of y when x is known, and the potential impact of the "purely real/imaginary" conditions. Ultimately, it is determined that the relationship between aHc, bHc, and c = xa + iyb can be used to find A and B as functions of x and y, but further progress requires additional information
  • #1
weetabixharry
111
0
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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  • #2
hi weetabixharry! :smile:
weetabixharry said:
(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 ?
 
  • #3
tiny-tim said:
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.
 
  • #4
weetabixharry said:
Yes, where I guess the coefficients of the linear combination are complex scalars.
From which it follows that y=0?
 
  • #5
haruspex said:
From which it follows that y=0?

Why?
 
  • #6
haruspex said:
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.
 
  • #7
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.
 
  • #8
haruspex said:
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
 
  • #9
Suppose you found a y and c =xa+iyb which satisfied all the conditions. Wouldn't 2y and c' =xa+i2yb also satisfy them?
 
  • #10
hi weetabixharry! :smile:

(just got up :zzz:)
weetabixharry said:
c = xa + iyb

that's right! :smile:
which is 1 equation in 2 unknowns (y and c)...

so what's the answer to :wink: … ?
weetabixharry said:
Given that I know x, can I deduce y?
 

What is the inner product of complex vectors?

The inner product of complex vectors is a mathematical operation that takes two complex vectors and produces a single complex number. It is also known as the dot product or scalar product.

How is the inner product of complex vectors calculated?

The inner product of complex vectors is calculated by multiplying the corresponding elements of the two vectors, adding them together, and then taking the complex conjugate of the result. This can be represented mathematically as (u, v) = u*v*, where u and v are complex vectors and * represents complex conjugation.

What is the significance of the inner product of complex vectors?

The inner product of complex vectors has many applications in mathematics and physics. It is used to define the length or magnitude of a vector, as well as the angle between two vectors. It also plays a key role in the definition of orthogonality and projection in vector spaces.

How is the inner product related to the magnitude and direction of a vector?

The magnitude of a complex vector can be calculated using the inner product as ||u|| = √(u, u). Similarly, the direction of a complex vector can be determined using the inner product as cosθ = (u, v)/||u|| ||v||, where θ is the angle between the two vectors.

Can the inner product of complex vectors be extended to higher dimensions?

Yes, the inner product of complex vectors can be extended to higher dimensions. In fact, the inner product can be defined for any vector space, as long as the vectors in that space satisfy certain properties. These properties include linearity, symmetry, and positive definiteness.

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