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

The conjugation of A and B does not change the fact that they are still complex numbers, and the multiplication is still done in the usual way.
  • #1
TheCanadian
367
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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?
 
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  • #2
TheCanadian said:
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.
TheCanadian said:
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.
 
  • #3
TheCanadian said:
## 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.
 
  • #4
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.
 
  • #5
phyzguy said:
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.
 
  • #6
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?
 
  • #7
If A and B are complex numbers, the notation has a clear, single meaning.
 
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  • #8
mfb said:
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?
 
  • #9
## E = AB - B^*A^*## is computed as ## E = AB - B^*A^*##.
 

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

1. What is the definition of a product of complex numbers?

A product of complex numbers is the result of multiplying two or more complex numbers together. It is similar to multiplying real numbers, but with the added consideration of imaginary numbers.

2. How do you multiply two complex numbers?

To multiply two complex numbers, you can use the FOIL method, which stands for First, Outer, Inner, and Last. This involves multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms. After multiplying, you can combine like terms to simplify the final product.

3. What is the difference between the product of two complex numbers and their sum?

The product of two complex numbers is the result of multiplication, while their sum is the result of addition. The product of two complex numbers is another complex number, while the sum of two complex numbers may or may not be a complex number.

4. Can the product of two complex numbers be a real number?

Yes, the product of two complex numbers can be a real number. This can happen when the imaginary components of the complex numbers are equal and opposite, resulting in a product of 0. In this case, the product is a real number.

5. How is the product of complex numbers used in real life?

The product of complex numbers has many applications in fields such as engineering, physics, and economics. It is often used in electrical engineering to calculate AC circuit currents and voltages, and in physics to represent electromagnetic waves. In economics, complex numbers are used to model and predict economic trends and behaviors.

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