## Suggestions for Complex Matrix Expansions

*Ideas for further contemplation from the blog post*

**Complex Matrix Expansions**.Consider the following properties that arise from the correspondence between a complex number $x + y\mathrm{i}$ and the matrix $\left[\begin{smallmatrix}x & -y\\y & x\end{smallmatrix}\right]$:

- If $z = 0$ then $\mathbf{Z} = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}.$ (zero matrix)
- If $z = 1$ then $\mathbf{Z} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}.$ (identity matrix)
- $z^{-1}$ corresponds to $\mathbf{Z}^{-1}.$ (matrix inverse)
- $z^\ast$ (complex conjugate) corresponds to $\mathbf{Z}^\intercal.$ (matrix transpose)
- $\det(\mathbf{Z}) = \lvert z\rvert^2. $

Try the following:

- Instead of expanding complex numbers into $2 \times 2$ real matrices, expand $m \times n$ complex matrices into $2m \times 2n$ real matrices.
- Find analogues of every one of the above-mentioned properties under this generalized expansion. Warning: the analogue of (5) is not obvious!
- Back then I had difficulty proving the analogue of (5) so I wrote a computer program to verify it by generating loads of randomized complex matrices and implementing arbitrary-precision arithmetic to multiply them.