<s>[INST] Null space of a hermitian matrix I have the following matrix:
$\begin{pmatrix}
4 &2-i  &-3i \\ 
2+i &1  &1-i \\ 
3i &1+i  &1 
\end{pmatrix}$
The following vector generates its null space:
$\begin{pmatrix}
-1\\ 
1+2i\\ 
1
\end{pmatrix}$
When I tried to find the null space, I wrote the original system as a symmetric real matrix of size $6\times 6$ by separating my matrix into its real and imaginary components:
$\begin{pmatrix}
4 &2  &0  &0  &1  &3 \\ 
2 &1  &1  &-1  &0  &1 \\ 
0 &1  &1  &-3  &-1  &0 \\ 
0 &-1  &-3  &4  &2  &0 \\ 
1 &0  &1  &2  &1  &1 \\ 
3 &1  &0  &0  &1  &1 
\end{pmatrix}$
The vector $\begin{pmatrix}
-1\\ 
1\\ 
1\\ 
0\\ 
2\\ 
0
\end{pmatrix}$ is supposed to be in its nullspace but wolframalpha tells me it's not:
https://www.wolframalpha.com/input/?i=%7B%7B4,2,0,0,1,3%7D,%7B2,1,1,-1,0,1%7D,%7B0,1,1,-3,-1,0%7D,%7B0,-1,-3,4,2,0%7D,%7B1,0,1,2,1,1%7D,%7B3,1,0,0,1,1%7D%7D*%7B-1,1,1,0,2,0%7D
I can't find where I went wrong. Could someone please help. Thanks in advance.
 [/INST] Wolfram tells me that $\left( \begin{array}{c}
0 \\ -2 \\ 0 \\ -1 \\ 1 \\ 1 \end{array} \right)$
is in the null space of the $6 \times 6$ matrix you've done there, which is similar to yours, but with the halves switched, and the wrong sign on the two.
Are you sure you have the right $6 \times 6$ matrix? Since the null space uses left-multiplication:
$$
\left( \begin{array}{ccc}
4 & 2-i & -3i \\
2+i & 1 & 1-i \\
3i & 1+i & 1 \end{array} \right)
\left( \begin{array}{c}
x_1 \\ x_2 \\ x_3 \end{array} \right)
= \left( \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right)
$$
shouldn't the rows match your matrix and not the columns?
This would give you:
$$
\left( \begin{array}{cccccc}
4 & 2 & 0 & 0 & -1 & -3 \\
2 & 1 & 1 & 1 & 0 & -1 \\
0 & 1 & 1 & 3 & 1 & 0 \\
0 & 1 & 3 & 4 & 2 & 0 \\
-1 & 0 & 1 & 2 & 1 & 1 \\
-3 & -1 & 0 & 0 & 1 & 1 \end{array} \right)
$$
...and, in fact, a quick bash into Wolfram reveals 
$\left( \begin{array}{c}
-1 \\ 1 \\ 1 \\ 0 \\ 2 \\ 0 \end{array} \right)$
to be in the null space.
I hope this is right, I have no idea what a hermitian matrix is, and I've never seen this approach used for anything before!
</s>