I'm not sure that this will make a lot of sense, but I guess that I'm trying to decide which course of study or action to persue.
I have a system of equations that seems to be linear, if the complex numbers are treated as scalars.
In other words, I have a system of equations with complex coefficients. I'm wondering how to go about solving these, and if linear algebra may somehow suffice. The variables are complex themselves, but I wonder whether scalar variables (instead of complex variables) would change anything.
The question/Summary
To make more sense of this question and to sum up what I'm looking for, I'd like to know the best way to solve a system of equations with complex coefficients and variables. Which branch(es) of math/science does this involve? An example showing how to solve a system of equations of this form would be particularly helpful, but not necessary.
An Example I'm interested In
I have a set of values, and I multiply them by complex values taken from the unit circle. So I may have a set${a,b,c}$ of complex values and I take this set and multiply it by $e^{i\pi d}$, which gives me three coefficients to use for three variables, say $x_1$, $x_2$, and $x_3$. I repeat this process several times (for different $d$ to create PART of several equations. So far, I can create euqations with these values, but they are complex multiples of one another.
These values (of the $x$'s) are added to another set of values that come from powers of complex numbers – this essentially ensures that the system of equations aren't "scalar" multiples of one another. I calculate corresponding sums of the variables. Then I have a system of equations.
What I'm trying to say is that I get a system of equations that aren't complex multiples of one another. I'm really thrilled at the possibility that I can solve this by linear algebra methods.
Best Answer
Yes: linear algebra works over any field, in particular over the complex numbers.
If your variables and coefficients are complex numbers, but the equations are linear on the unknowns, then the usual methods of complex numbers work, exactly as they do over the real numbers.
Rather than me invent a system of equations that may not match what you are looking for, perhaps you can post an example yourself, both in the "complex variable" and "scalar variable" case (I confess to being somewhat confused about what you mean by the latter).