What are the "other" Hadamard matrices?

Question

The Pauli matrices $$ X = \begin{pmatrix}0&1\\1&0\end{pmatrix}, Y=\begin{pmatrix}0&-i\\i &0\end{pmatrix},\,\text{and}\, Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix} $$ can be used to construct the Hadamard gate $$ H=\frac{1}{\sqrt{2}}(X+Z)=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\1&-1\end{pmatrix}. $$ $H$ is Hermitian and two other Hermitian matrices arise when doing a similar computation with the $Y$ matrix: $$ \frac{1}{\sqrt{2}}(X+Y)=\frac{1}{\sqrt{2}}\begin{pmatrix}0 & 1-i \\ 1+i & 0\end{pmatrix}\quad\text{and}\quad\frac{1}{\sqrt{2}}(Y+Z)\begin{pmatrix}1 & -i \\ i & -1\end{pmatrix} $$ Do these matrices have some names and if yes, do there exist known properties about them?

Answer 1

Realize that an arbitrary rotation around the axis $\mathbf{n}$ is given by $R_\mathbf{n}=\cos(\alpha/2)I-i\sin(\alpha/2)\hat{\mathbf{n}}\cdot\mathbf{\sigma}$ and an arbitrary unitary operator can be written as $U=\exp{(i\gamma)}R_\mathbf{n}$ with $\gamma$ some phase factor. Thus, in general, any operations on the qubit can be seen as a rotation with some phase factor. Nevertheless, in the literature only the Hadamard, X,Y,Z and the phase shift gate are usually mentioned since these gates are conceptually the most practical to use on a single qubit.