A mirror, not a square

Stand the matrix on its corner, and the condition $H = H^{\dagger}$ becomes a literal mirror reflection.

Full paragraph (abstract translation)

The matrix representation of a complex number carries a second hidden symmetry. In the standard layout, an $N \times N$ matrix is a square whose diagonal runs from the top-left corner to the bottom-right, and the Hermiticity condition $H = H^{\dagger}$ looks like a purely algebraic statement.

Stand the matrix on its corner, as a rhombus — so that the diagonal becomes vertical. The real diagonal entries end up exactly on the axis of symmetry; the entry $(i, j)$ and its conjugate partner $(j, i)$ land in literally mirror-symmetric positions across that axis. Hermiticity, invisible in the square layout, becomes a geometric reflection visible to the naked eye.

Why this matters for the talk's storyline

• A geometric interpretation of Hermiticity is propaedeutic to $\mathbb{Z}[\omega]$: when the lattice swaps the square for the hexagon (slide 5), this "stand it on its corner" trick will become the default coordinate system.
• It is also preparation for isotropy of error quantization in GQ128: there, too, the standard IEEE 754 "square" gets in the way of seeing the real symmetry.