Dot product as matrix products?

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Discussion Overview

The discussion revolves around the conventions used for the dot product of vectors, particularly in the context of real and complex vectors. Participants explore the implications of different notations and the resulting mathematical structures, including scalars and matrices.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants propose defining the dot product as dot(u,v) = transpose(u)v instead of transpose(v)u.
  • Others question the difference between these two definitions.
  • One participant notes that the distinction becomes significant when using Hermitian conjugates with complex vectors, highlighting different conventions used by physicists and mathematicians.
  • It is mentioned that one way generates a scalar while the other generates an NxN matrix, depending on whether the vectors are treated as row or column vectors.
  • A participant challenges the logic of the previous responses, asserting that if both u and v are Nx1 vectors, both forms yield a single component 1x1 matrix.
  • Another participant clarifies that if u and v are column vectors, u^Tv results in a scalar, while uv^T results in a matrix, emphasizing the importance of distinguishing between the two forms.
  • There is a recognition that the question pertains to the difference between u^Tv and v^Tu, with the conclusion that for real numbers, there is no difference, but for complex numbers, one is the complex conjugate of the other.

Areas of Agreement / Disagreement

Participants express differing views on the conventions for the dot product, particularly in the context of complex vectors. There is no consensus on which convention is superior, and the discussion remains unresolved regarding the implications of these conventions.

Contextual Notes

The discussion highlights the dependence on definitions and conventions in mathematical notation, particularly between different fields such as physics and mathematics. The implications of using Hermitian conjugates versus transposes are also noted, but not fully resolved.

tgt
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Why not make dot(u,v)=transpose(u)v rather than transpose(v)u?
 
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What's the difference?
 
At least difference emerges when transpose is replaced with Hermitian conjugate, when complex vectors are used. Physicist use the convention

[tex] (u|v) = u^{\dagger} v[/tex]

and IMO it is lot better than the mathematicians' convention

[tex] (u|v) = v^{\dagger} u[/tex]

When something is done in a dumb way, the reason is usually "for historical reasons". I guess that's the answer to the OP question this time too.
 
Hurkyl said:
What's the difference?
One way generates a scalar and the other way generates an NxN matrix. Which is which depends on whether the vector is a 1xN row vector or a Nx1 column vector.
 
D H said:
Hurkyl said:
What's the difference?
One way generates a scalar and the other way generates an NxN matrix. Which is which depends on whether the vector is a 1xN row vector or a Nx1 column vector.

This response is not logical!

If we assume [itex]u[/itex] and [itex]v[/itex] to be Nx1 vectors, then both [itex]u^Tv[/itex] and [itex]v^Tu[/itex] give a single component 1x1 matrix.

tgt did not ask about why to use Nx1 or 1xN matrices, so it is better not to start switching between them now.

The truth is that there are two different conventions for complex inner products, and they are

[tex] (u|v) = \sum_{k=1}^N u^*_k v_k[/tex]

and

[tex] (u|v) = \sum_{k=1}^N u_k v^*_k[/tex]

so I thought it would be natural to guess that the original question was related to this issue.
 
jostpuur said:
At least difference emerges when transpose is replaced with Hermitian conjugate, when complex vectors are used.
I see now.
 
D H said:
One way generates a scalar and the other way generates an NxN matrix. Which is which depends on whether the vector is a 1xN row vector or a Nx1 column vector.
No, you have misread. If u and v are column vectors (most common convention), then uTv is a scalar and uvT is a matrix.

But the question was about the difference between uTv and vTu, both of which are scalars. And the answer is that if the vectors are over the real numbers, there is no difference and if the vectors are over the complex numbers, one is the complex conjugate of the other. In the latter case, which we use as inner product is a matter of convention.
 
Yes, I misread the OP as uTv versus uvT, as opposed to uTv versus vTu.
 

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