Solve equations real and imaginary part separately

For my system of equations, the procedure described in Solving complex equations of using Reduce works no more. How can I separate the real and imaginary part of the equations? Because then I could use Solve[equations, vars, Reals]. Nevertheless I hope for a simpler way to overcome this issue.

Example

Vector = {v1, v2, v3, v4};
Matrix = {{c11, c12, c13, c14},
{c21, c22, c23, c24},
{c31, c32, c33, c34},
{c41, c42, c43, c44}};
Reduce[Table[0 == Sum[Matrix[[r, k]] Vector[[k]], {k, 4}], {r, 4}] &&
Element[{v1Real, v1Complex, v2Real, v2Complex, v3Real, v3Complex, v4Real, v4Complex}, Reals],
{v1Real, v1Complex, v2Real, v2Complex, v3Real, v3Complex, v4Real, v4Complex}] /.
{v1 -> v1Real + I v1Complex, v2 -> v2Real + I v2Complex, v3 -> v3Real + I v3Complex, v4 -> v4Real + I v4Complex}
• We could likely provide something more useful to you if on;y you'd explicitly mention what your "system of equations" are. – J. M. will be back soon May 2 '13 at 12:05
• The way you defined the equations you can only expect trivial solutions. Try to check eq = (# == 0) & /@ (matrix . vector);Solve[eq, vector, Complexes]. Also be careful with the fact that initiating symbols with capital letters are avoided in Mathematica. – PlatoManiac May 2 '13 at 12:41
• I want to introduce some symmetry properties into the vector as in my case the vector represents the density matrix, i.e. Complexes is no intuitive option! – strpeter May 2 '13 at 14:10
• It's a linear system. You can use LinearSolve. Then, if you must, separate the symbolic solution into real and imaginary parts using Re and Im. – Daniel Lichtblau May 2 '13 at 14:44