Complex numbers in my system of equations

Question

This is for an optimization problem? I haven't dealt with complex numbers(although I know the eigenvalues are going to give me a neutral circular node)

$\begin{bmatrix}-\sqrt{2}i & 2\\ -2 & -\sqrt{2}i\end{bmatrix}\begin{bmatrix}x_1 \\x_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}$

How to solve for $x_1$ & $x_2$?

Thanks

Answer 1

Hint: If you do the matrix multiplication, you'll get a $2\times1$ matrix. The top and bottom entries must equal the top and bottom entries on the other side of the equals sign, so you get a system of equations you can solve.

Answer 2

Since the system is homogeneous, the unique solution is the ordered pair $(x_1,x_2)=(0,0)$.

Answer 3

This matrix is invertible. Check the determinant: $(-\sqrt{2}i \cdot -\sqrt{2}i) - (-2 \cdot 2) = -2 + 4 = 2$. So the nullspace is trivial, and so $x_1 = x_2 = 0$.