What is the Hermitian Conjugate of 5+6i?

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The Hermitian conjugate of a complex number, such as 5+6i, is essentially its complex conjugate, which is 5-6i. The term "Hermitian conjugate" is primarily associated with matrices and refers to the conjugate transpose of a matrix with complex entries. When considering a complex number as a 1x1 matrix, its Hermitian conjugate aligns with its complex conjugate. In mathematical terms, the Hermitian conjugate of an operator is defined through the relationship <x,Ay> = <A*x,y>, highlighting its role as an adjoint. This concept is particularly significant in quantum mechanics, especially for infinite-dimensional operators.
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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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