Split complex system of equations into two real systems

Question

Suppose I have a complex system of equations in 3 unknowns, like this one:

$$ \pmatrix{ 40 & -20 & 0\\ -20 & 20-20j & 30+10j\\ 4 & -5 & 1 } \pmatrix{ x_1+j x_2\\ y_1+j y_2\\ z_1+j z_2\\ }= \pmatrix{ 10\\ 0\\ 0\\ } $$

Here are solutions:

I want to split the matrix with complex numbers to two matrices with real numbers. However the imaginary matrix would only have one equation:

$$-20y_2+10z_2=0$$

What am I doing wrong?

Or is it possible to split such a system to two systems in the first place? (two "real" systems?)

Answer 1

Also $X$, $Y$ and $Z$ should be split into real and imaginary parts.

The system becomes $$ \begin{cases} 40(x_1+jx_2)-20(y_1+jy_2)=10\\ -20(x_1+jx_2)+(20-20j)(y_1+jy_2)+(30+10j)(z_1+jz_2)=0\\ 4(x_1+jx_2)-5(y_1+jy_2)+(z_1+jz_2)=0 \end{cases} $$ so you get $$ \begin{cases} 40x_1-20y_1=10\\ 40x_1-20y_2=0\\ -20x_1+20y_1+20y_2+30z_1-10z_2=0\\ -20x_2-20y_1+20y_2+10z_1+30z_2=0\\ 4x_1-5y_1+z_1=0\\ 4x_2-5y_2+z_2=0 \end{cases} $$ which are six equations in six unknowns.

However, there's no need to do this, because complex numbers are as well behaved as the real numbers, when solving linear systems.

Answer 2

Yes, just separate the real and imaginary parts all the way through.

In terms of arithmetic complex-numbers are isomorphic with 2-D vectors. However, this would give you twice as much work to do.

The hardness of the equation(s) is because they involve a 3 by 3 matrix, and 3 unknown quantities, and this does not simplify by separating real and imaginary components (parts).