My quantum mechanics professor was discussing the properties of Pauli matrices, their being both Hermitian and unitary. Then he made a remark that it is not possible to find three $n \times n$ matrices, where $n > 2$, that are simultaneously Hermitian and unitary. Can someone please explain or give a hint as to why this has to be true?
2026-03-27 06:09:49.1774591789
Matrices that are simultaneously Hermitian and unitary
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Three of the four Dirac matrices $$ \gamma^1=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&-1&0&0\\-1&0&0&0\end{matrix}\right)\,,\quad \gamma^2=\left(\begin{matrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{matrix}\right)\,,\quad \gamma^3=\left(\begin{matrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{matrix}\right) $$ are obviously anti-Hermitian and unitary. Therefore, their multiples with $\pm i$ are Hermitian and unitary.