Conjugate of a matrix and of a function

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Discussion Overview

The discussion revolves around the concept of conjugates in the context of matrices and functions, particularly in relation to linear vector spaces (LVS) and quantum mechanics (QM). Participants explore the differences in how conjugates are defined and understood for functions versus matrices, and the implications of these definitions in various mathematical contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that the conjugate of a function in a LVS is consistent, while the conjugate of a matrix can vary, expressing confusion about the reliance on complex numbers for the importance of conjugates in functional spaces.
  • Another participant clarifies that the conjugate of a matrix involves replacing its entries with their complex conjugates, and if the matrix has all real entries, the conjugate is the matrix itself.
  • There is a suggestion that the original poster may be conflating "conjugate" with "conjugate transpose" or "Hermitian transpose," which involves transposing the matrix and taking the conjugate of each entry.
  • A question is raised regarding the representation of functions as matrices and why functions do not have a transpose operation analogous to matrices.
  • A participant references operator functions in quantum mechanics, suggesting they are similar to matrices.
  • Another participant introduces the concept of the adjoint of a linear transformation, explaining its relationship to the conjugate transpose of a matrix in the context of inner-product spaces.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between functions and matrices, particularly regarding the definitions and implications of conjugates and transposes. The discussion remains unresolved, with multiple perspectives on the nature of these mathematical concepts.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the definitions of conjugates and transposes, as well as the specific contexts in which these terms are applied. The relationship between functions and matrices is not fully explored, leaving some questions open.

DeepSeeded
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Hello,

Working without complex numbers a conjugate of any function in a LVS is always the same thing. A conjugate of any matrix in a LVS is very often not the same thing though. I am just confused as to why functional spaces rely on complex numbers for the conjugate to have any importance and a matrix does not.
 
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Where did you see that? The "conjugate" of a matrix is just the matrix with the entries replaced by there complex conjugates. If M is a matrix with all real entries then the conjugate of M is just M itself.

You may be confusing "conjugate" with the "conjugate transpose" or "Hermitian transpose" of a matrix: swap rows and columns and take the conjugate of each entry. Of course, if M has all real entries, it "conjugate transpose" is just its transpose.
 
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HallsofIvy said:
Where did you see that? The "conjugate" of a matrix is just the matrix with the entries replaced by there complex conjugates. If M is a matrix with all real entries then the conjugate of M is just M itself.

You may be confusing "conjugate" with the "conjugate transpose" or "Hermitian transpose" of a matrix: swap rows and columns and take the conjugate of each entry. Of course, if M has all real entries, it "conjugate transpose" is just its transpose.

So guess my question is if functions are a different represenation of a matrix why is there no option to transpose a function?
 
In what sense is a function a "different representation of a matrix"? Are you talking about representing linear functions represented by a matrix?
 
Last edited by a moderator:
In my QM class Operator functions are said to be like a matrix.
 
I don't fully understand your question, but maybe you'd like to hear about the adjoint of a linear transformation.

Let V,W be inner-product spaces, let T\in L(V,W) be a linear transformation, and T^*\in L(W,V) its adjoint. This means that \langle Tv,w \rangle=\langle v,T^*w \rangle for all v\in V,w\in W. Then, the matrix of T^* with respect to orthonormal bases of V and W is just the conjugate transpose of the matrix of T with respect to these bases. As mentioned earlier, the conjugate transpose of a matrix is just the transpose (interchange rows and colums) of the matrix with all entries replaced by their complex conjugates.
 

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