What is the Hermitian Conjugate of 5+6i?

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Discussion Overview

The discussion centers around the concept of the Hermitian conjugate, particularly in relation to the complex number 5+6i. Participants explore the definitions and applications of the Hermitian conjugate in the context of complex numbers and matrices, touching on theoretical and conceptual aspects.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions what the Hermitian conjugate of the complex number 5+6i is.
  • Another participant asserts that the term "Hermitian conjugate" typically refers to the conjugate transpose of matrices with complex entries, suggesting that the only conjugate for a complex number is the regular complex conjugate.
  • A different viewpoint suggests that while "Hermitian conjugate" is generally used for matrices, if one considers a complex number as a "1 by 1 matrix," its Hermitian conjugate would be its complex conjugate, which is 5-6i.
  • One participant elaborates on the definition of the Hermitian conjugate as the adjoint of an operator, noting that for finite-dimensional operators, it equates to the transpose conjugate, but emphasizes its significance in quantum mechanics with infinite-dimensional operators.

Areas of Agreement / Disagreement

Participants express differing views on the application of the term "Hermitian conjugate" to complex numbers versus matrices, indicating a lack of consensus on its usage in this context.

Contextual Notes

Some participants highlight the distinction between the Hermitian conjugate as it applies to matrices and its relevance to complex numbers, but the discussion does not resolve the implications of this distinction.

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What is the Hermitian conjugate of a complex #, say, 5+6i??
 
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As far as I know the name Hermitian conjugate is alternate name for the conjugate transpose of a matrix with complex entries. I think the only type of conjugate for a complex number is the regular one:

<br /> \overline{5-6 i} = 5 + 6i<br />
 
"Hermitian conjugate" is usually used for matrices, not numbers. However, if you think of a+ bi as a "1 by 1 matrix" then its Hermitian conjugate is just its complex conjugate, a- bi.
 
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statdad said:
As far as I know the name Hermitian conjugate is alternate name for the conjugate transpose of a matrix with complex entries.

Actually the Hermitian conjugate A* of an operator A is defined by

<x,Ay> = <A*x,y>

in other words the Hermitian conjugate is what mathematicians call an adjoint. It turns out that for finite dimensional operators (matrices) the Hermitian conjugate is simply equal to the transpose conjugate, as you state, but the more general definition is highly important in quantum mechanics, where the momentum operator is infinite-dimensional.
 

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