I find two different rotation matrices are mapped to a single quaternion.
$ \begin{pmatrix} -1 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ 0 & -\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \end{pmatrix} $
$ \begin{pmatrix} -1 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ 0 & \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \end{pmatrix} $
They are both mapped to a quaternion:
$w: 0$
$x: 0$
$y: 0.9238795325112867$
$z: 0.38268343236508984$
Is it a problem relate to my matrix-quaternion conversion program?