How Do You Compute the Expression E = AB - B^*A^* with Complex Numbers?

[TheCanadian]

If I have 2 complex numbers, A and B, what is the correct way to evaluate this expression:

$E = AB - B^*A^*$

I was under the impression that when taking the product of complex numbers, you always conjugate one factor, but in this instance, it is quite important which one is conjugated, no? For example, is the correct way to compute E:

$E = AB^* - BA^*$ or $E = A^*B- B^*A$ or another method?

 

[blue_leaf77]

If I have 2 complex numbers, A and B, what is the correct way to evaluate this expression:

$E = AB - B^*A^*$

The correct way to calculate $E$ is to calculate the RHS as it is.

I was under the impression that when taking the product of complex numbers, you always conjugate one factor,

I was under the impression that you mixed the ordinary product between complex numbers with an inner product on a complex vector space.

 

[mfb]

$E = AB^* - BA^*$ or $E = A^*B- B^*A$ or another method?

Why do you want to change the expression?

You evaluate AB.
You evaluate B* and A* and multiply them to get B*A*.
You subtract both.

Using some rules for complex numbers you can save a bit of time, but that is completely optional.

 

[phyzguy]

There is no "correct way". You can multiply complex numbers without conjugating either one, or you can conjugate one of them, or you can conjugate both of them. It depends what you are doing. Each of the three expressions you have written for E is different from the other two. For example, if:
A = 2+3i
B = 4+7i
Then your first expression for E is 52i, your second expression for E is -4i, and your third expression for E is 4i.

 

[TheCanadian]

There is no "correct way". You can multiply complex numbers without conjugating either one, or you can conjugate one of them, or you can conjugate both of them. It depends what you are doing. Each of the three expressions you have written for E is different from the other two. For example, if:
A = 2+3i
B = 4+7i
Then your first expression for E is 52i, your second expression for E is -4i, and your third expression for E is 4i.

Thank you for the responses. The main reason I was asking was because I saw the initial expression I posted in a paper but wasn't exactly sure how the author intended the expression to be evaluated.

 

[blue_leaf77]

Then you have to know in which context the equation is presented, e.g. what do those alphabets symbolize, are they scalars, vectors, or linear transformation?

 

[mfb]

If A and B are complex numbers, the notation has a clear, single meaning.

 

[TheCanadian]

If A and B are complex numbers, the notation has a clear, single meaning.

Ahh yes, I think I confused the exact computation because A and B are the complex amplitudes (scalar functions dependent on the spatial and temporal variables) of complex vectors, but are not vectors themselves. And so in such a case, just to confirm, the expression E above would be computed as initially stated?

 

[mfb]

$E = AB - B^*A^*$ is computed as $E = AB - B^*A^*$.