Should a Pauli $X$ matrix equal the identity matrix to be unitary?


My understanding is that any unitary matrix must have its inverse be equal to its conjugate transpose.

Looking at the pauli x gate as shown here: $$\begin{bmatrix}0&1\\1&0\end{bmatrix}$$

It is its own inverse which is equal, of course, to its own conjugate transpose.

However, isn't it also true that neither of these form an identity matrix? And isn't this a requirement for being considered unitary?


Posted 2019-09-09T18:40:51.597

Reputation: 285



No, the matrix, its inverse or its conjugate transpose don't have to equal the identity matrix to be unitary (that would make the class of unitary matrices rather restricted).

The definition of unitary matrix is that its conjugate transpose should equal its inverse, i.e., $U U^\dagger = U^\dagger U = I$. You can check that this is indeed the case for the Pauli X matrix.

Mariia Mykhailova

Posted 2019-09-09T18:40:51.597

Reputation: 6 616

So in UU†=U†U=I, how does the =I make sense if the conjugate transpose does not have to equal the id matrix? sorry to be dense, appreciate your breaking it down here. – neutrino – 2019-09-09T19:35:42.337

3The product of $U$ with its conjugate transpose $U^\dagger$ should equal the identity matrix. The conjugate transpose $U^\dagger$ does not need to equal $I$. Matrix multiplication is not, in general, commutative. – Mark S – 2019-09-09T21:08:29.797