One of the basic gates in quantum computing is a Hadamard gate, action of which is given by the following matrix

H $=12$

$11$

$11$

$.$

$($a$)$ Find the output of the Hadamard gates if the input qubit is in $|\backslash$pm z$,|\backslash$pm x$$ or $|\backslash$pm y$$ states.

$($b$)$ Find the eigenvectors and eigenvalues of H