System of linear equations with complex coefficients

1. Jul 29, 2009

Qubix

1. You have a system of equations of the following form:
x + iy = 0
-ix + z = 0
y - z = 0

or

-2Sqr(5)x - iy = 0
ix - 2Sqr(5)y + 2iz = 0
-2iy - 2Sqr(5)z = 0

2. What is the general way in which I can solve such a system? I've tried Kramer's Rule, but it does not seem to work since Dx, Dy, Dz all give zero.

2. Jul 29, 2009

rock.freak667

Quite easily it can be seen that (0,0,0) is a solution here

What you can do is form an augmented matrix and row reduce.

or take the first equation x+iy=0 and put it such that x= -iy and put that into the other equations that have 'x' in it. You will now have two equations with two unknowns.

Do the same with the other set of equations.

3. Jul 30, 2009

HallsofIvy

. From -ix+ z= 0, ix= z so x= -iz. We also have x= -iy so -iy= -iz or y= z. The last equation is y- z= 0 which says y= z also. That's why all the determinant are 0: this system has an infinite number of solutions. You can choose z to be anything you want and write x and y in terms of z.

4. Jul 31, 2009