# Rows and columns in quantum-gate matrices read the same - why?

I have noticed that for all the matrices representing quantum gates, if we read rows left-to-right and top to bottom, the read the same as columns top to bottom left to right.

Example:

\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1& 0 \end{pmatrix} (CNOT gate), if we read the values in from the first row, we see 1 0 0 0. Similarly if we read left-most column top to bottom. Second row reads 0 1 0 0, just like 2nd left column etc.

I believe this is a property of all matrices representing quantum gates, but cannot explain why exactly this is the case (and if it is, in fact, guaranteed for every quantum gate?)

• This is because these gates are Hermitian, not because they're unitary! – Yuval Filmus Jul 7 '18 at 15:10
• Thanks @YuvalFilmus! Would you consider wrapping it in an answer, and I shall change the question accordingly? – 3yakuya Jul 7 '18 at 20:02

$$Y = \begin{bmatrix} 0&-i\\i&0\end{bmatrix}$$
$$\frac{1-i}{2} \begin{bmatrix} 1&-i\\1&i\end{bmatrix}$$