# Hermitian Conjugate

1. Oct 28, 2008

### yzou_ua

What is the Hermitian conjugate of a complex #, say, 5+6i??

2. Oct 28, 2008

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:

$$\overline{5-6 i} = 5 + 6i$$

3. Oct 29, 2008

### HallsofIvy

Staff Emeritus
"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.

Last edited: Oct 30, 2008
4. Oct 29, 2008

### Crosson

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.