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(x, y, z) = var('theta_x, theta_y, theta_z')
(sx, sy, sz) = (sin(x), sin(y), sin(z))
(cx, cy, cz) = (cos(x), cos(y), cos(z))

Rx = Matrix([
    [1,      0,       0],
    [0, cos(x), -sin(x)],
    [0, sin(x),  cos(x)]
])

Ry = Matrix([
    [ cos(y), 0, sin(y)],
    [      0, 1,      0],
    [-sin(y), 0, cos(y)]
])

Rz = Matrix([
    [cos(z), -sin(z), 0],
    [sin(z),  cos(z), 0],
    [     0,       0, 1]
])

show( '$R_x=$', Rx, '$, R_y=$', Ry, '$, R_z=$', Rz )
show( '$R_x R_y R_z=$', Rx * Ry * Rz )
Rx=R_x= (1000cos⁡(θx)−sin⁡(θx)0sin⁡(θx)cos⁡(θx))\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & \cos\left(\theta_{x}\right) & -\sin\left(\theta_{x}\right) \\ 0 & \sin\left(\theta_{x}\right) & \cos\left(\theta_{x}\right) \end{array}\right) ,Ry=, R_y= (cos⁡(θy)0sin⁡(θy)010−sin⁡(θy)0cos⁡(θy))\displaystyle \left(\begin{array}{rrr} \cos\left(\theta_{y}\right) & 0 & \sin\left(\theta_{y}\right) \\ 0 & 1 & 0 \\ -\sin\left(\theta_{y}\right) & 0 & \cos\left(\theta_{y}\right) \end{array}\right) ,Rz=, R_z= (cos⁡(θz)−sin⁡(θz)0sin⁡(θz)cos⁡(θz)0001)\displaystyle \left(\begin{array}{rrr} \cos\left(\theta_{z}\right) & -\sin\left(\theta_{z}\right) & 0 \\ \sin\left(\theta_{z}\right) & \cos\left(\theta_{z}\right) & 0 \\ 0 & 0 & 1 \end{array}\right)
RxRyRz=R_x R_y R_z= (cos⁡(θy)cos⁡(θz)−cos⁡(θy)sin⁡(θz)sin⁡(θy)cos⁡(θz)sin⁡(θx)sin⁡(θy)+cos⁡(θx)sin⁡(θz)−sin⁡(θx)sin⁡(θy)sin⁡(θz)+cos⁡(θx)cos⁡(θz)−cos⁡(θy)sin⁡(θx)−cos⁡(θx)cos⁡(θz)sin⁡(θy)+sin⁡(θx)sin⁡(θz)cos⁡(θx)sin⁡(θy)sin⁡(θz)+cos⁡(θz)sin⁡(θx)cos⁡(θx)cos⁡(θy))\displaystyle \left(\begin{array}{rrr} \cos\left(\theta_{y}\right) \cos\left(\theta_{z}\right) & -\cos\left(\theta_{y}\right) \sin\left(\theta_{z}\right) & \sin\left(\theta_{y}\right) \\ \cos\left(\theta_{z}\right) \sin\left(\theta_{x}\right) \sin\left(\theta_{y}\right) + \cos\left(\theta_{x}\right) \sin\left(\theta_{z}\right) & -\sin\left(\theta_{x}\right) \sin\left(\theta_{y}\right) \sin\left(\theta_{z}\right) + \cos\left(\theta_{x}\right) \cos\left(\theta_{z}\right) & -\cos\left(\theta_{y}\right) \sin\left(\theta_{x}\right) \\ -\cos\left(\theta_{x}\right) \cos\left(\theta_{z}\right) \sin\left(\theta_{y}\right) + \sin\left(\theta_{x}\right) \sin\left(\theta_{z}\right) & \cos\left(\theta_{x}\right) \sin\left(\theta_{y}\right) \sin\left(\theta_{z}\right) + \cos\left(\theta_{z}\right) \sin\left(\theta_{x}\right) & \cos\left(\theta_{x}\right) \cos\left(\theta_{y}\right) \end{array}\right)

rXyz = Rx * Ry * Rz
rZyx = Rz * Ry * Rx

eberlyXyz = Matrix([
    [             cy * cz,              -(cy * sz),            sy    ],
    [   cz * sx * sy + cx * sz,    cx * cz - sx * sy * sz, -(cy * sx)],
    [-(cx * cz * sy) + (sx * sz),  cz * sx + cx * sy * sz,   cx * cy ]
])

eberlyZyx = Matrix([
    [cy*cz, (cz*sx*sy) - (cx*sz),  cx*cz*sy + sx*sz   ],
    [cy*sz,      cx*cz + sx*sy*sz, -(cz*sx) + cx*sy*sz],
    [ -sy,           cy*sx,               cx*cy       ]
])

unreal = Matrix([
    [            cy * cz,               cy * sz,            sy   ],
    [  sx * sy * cz - cx * sz,   sx * sy * sz + cx * cz, -sx * cy],
    [-(cx * sy * cz + sx * sz),  cz * sx - cx * sy * sz,  cx * cy]
])

show( eberlyXyz - rXyz )
show( eberlyZyx - rZyx )
show( unreal - rXyz )

#
#unreal = Matrix([
#    [cy*cz, sx*sy*cz - cx*sz, -(cx*sy*cz + sx*sz)  ],
#    [cy*sz, sx*sy*sz + cx*cz,      cz*sx - cx*sy*sz],
#    [  sy,       -(sx*cy),             cx*cy       ]
#])
#
#show( unreal - rZyx )
(000000000)\displaystyle \left(\begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)
(000000000)\displaystyle \left(\begin{array}{rrr} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)
(02 cos⁡(θy)sin⁡(θz)0−2 cos⁡(θx)sin⁡(θz)2 sin⁡(θx)sin⁡(θy)sin⁡(θz)0−2 sin⁡(θx)sin⁡(θz)−2 cos⁡(θx)sin⁡(θy)sin⁡(θz)0)\displaystyle \left(\begin{array}{rrr} 0 & 2 \, \cos\left(\theta_{y}\right) \sin\left(\theta_{z}\right) & 0 \\ -2 \, \cos\left(\theta_{x}\right) \sin\left(\theta_{z}\right) & 2 \, \sin\left(\theta_{x}\right) \sin\left(\theta_{y}\right) \sin\left(\theta_{z}\right) & 0 \\ -2 \, \sin\left(\theta_{x}\right) \sin\left(\theta_{z}\right) & -2 \, \cos\left(\theta_{x}\right) \sin\left(\theta_{y}\right) \sin\left(\theta_{z}\right) & 0 \end{array}\right)
rXyzFlipz = Rx * Ry * Rz.inverse()
show( unreal - rXyzFlipz )
(−(1cos⁡(θz)−sin⁡(θz)2(sin⁡(θz)2cos⁡(θz)+cos⁡(θz))cos⁡(θz)2)cos⁡(θy)+cos⁡(θy)cos⁡(θz)cos⁡(θy)sin⁡(θz)−cos⁡(θy)sin⁡(θz)(sin⁡(θz)2cos⁡(θz)+cos⁡(θz))cos⁡(θz)0−(1cos⁡(θz)−sin⁡(θz)2(sin⁡(θz)2cos⁡(θz)+cos⁡(θz))cos⁡(θz)2)sin⁡(θx)sin⁡(θy)+cos⁡(θz)sin⁡(θx)sin⁡(θy)−cos⁡(θx)sin⁡(θz)+cos⁡(θx)sin⁡(θz)(sin⁡(θz)2cos⁡(θz)+cos⁡(θz))cos⁡(θz)sin⁡(θx)sin⁡(θy)sin⁡(θz)+cos⁡(θx)cos⁡(θz)−sin⁡(θx)sin⁡(θy)sin⁡(θz)(sin⁡(θz)2cos⁡(θz)+cos⁡(θz))cos⁡(θz)−cos⁡(θx)sin⁡(θz)2cos⁡(θz)+cos⁡(θz)0(1cos⁡(θz)−sin⁡(θz)2(sin⁡(θz)2cos⁡(θz)+cos⁡(θz))cos⁡(θz)2)cos⁡(θx)sin⁡(θy)−cos⁡(θx)cos⁡(θz)sin⁡(θy)−sin⁡(θx)sin⁡(θz)+sin⁡(θx)sin⁡(θz)(sin⁡(θz)2cos⁡(θz)+cos⁡(θz))cos⁡(θz)−cos⁡(θx)sin⁡(θy)sin⁡(θz)+cos⁡(θz)sin⁡(θx)+cos⁡(θx)sin⁡(θy)sin⁡(θz)(sin⁡(θz)2cos⁡(θz)+cos⁡(θz))cos⁡(θz)−sin⁡(θx)sin⁡(θz)2cos⁡(θz)+cos⁡(θz)0)\displaystyle \left(\begin{array}{rrr} -{\left(\frac{1}{\cos\left(\theta_{z}\right)} - \frac{\sin\left(\theta_{z}\right)^{2}}{{\left(\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)\right)} \cos\left(\theta_{z}\right)^{2}}\right)} \cos\left(\theta_{y}\right) + \cos\left(\theta_{y}\right) \cos\left(\theta_{z}\right) & \cos\left(\theta_{y}\right) \sin\left(\theta_{z}\right) - \frac{\cos\left(\theta_{y}\right) \sin\left(\theta_{z}\right)}{{\left(\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)\right)} \cos\left(\theta_{z}\right)} & 0 \\ -{\left(\frac{1}{\cos\left(\theta_{z}\right)} - \frac{\sin\left(\theta_{z}\right)^{2}}{{\left(\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)\right)} \cos\left(\theta_{z}\right)^{2}}\right)} \sin\left(\theta_{x}\right) \sin\left(\theta_{y}\right) + \cos\left(\theta_{z}\right) \sin\left(\theta_{x}\right) \sin\left(\theta_{y}\right) - \cos\left(\theta_{x}\right) \sin\left(\theta_{z}\right) + \frac{\cos\left(\theta_{x}\right) \sin\left(\theta_{z}\right)}{{\left(\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)\right)} \cos\left(\theta_{z}\right)} & \sin\left(\theta_{x}\right) \sin\left(\theta_{y}\right) \sin\left(\theta_{z}\right) + \cos\left(\theta_{x}\right) \cos\left(\theta_{z}\right) - \frac{\sin\left(\theta_{x}\right) \sin\left(\theta_{y}\right) \sin\left(\theta_{z}\right)}{{\left(\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)\right)} \cos\left(\theta_{z}\right)} - \frac{\cos\left(\theta_{x}\right)}{\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)} & 0 \\ {\left(\frac{1}{\cos\left(\theta_{z}\right)} - \frac{\sin\left(\theta_{z}\right)^{2}}{{\left(\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)\right)} \cos\left(\theta_{z}\right)^{2}}\right)} \cos\left(\theta_{x}\right) \sin\left(\theta_{y}\right) - \cos\left(\theta_{x}\right) \cos\left(\theta_{z}\right) \sin\left(\theta_{y}\right) - \sin\left(\theta_{x}\right) \sin\left(\theta_{z}\right) + \frac{\sin\left(\theta_{x}\right) \sin\left(\theta_{z}\right)}{{\left(\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)\right)} \cos\left(\theta_{z}\right)} & -\cos\left(\theta_{x}\right) \sin\left(\theta_{y}\right) \sin\left(\theta_{z}\right) + \cos\left(\theta_{z}\right) \sin\left(\theta_{x}\right) + \frac{\cos\left(\theta_{x}\right) \sin\left(\theta_{y}\right) \sin\left(\theta_{z}\right)}{{\left(\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)\right)} \cos\left(\theta_{z}\right)} - \frac{\sin\left(\theta_{x}\right)}{\frac{\sin\left(\theta_{z}\right)^{2}}{\cos\left(\theta_{z}\right)} + \cos\left(\theta_{z}\right)} & 0 \end{array}\right)

def QuatFromMatrix( m ):
    # wxyz, or rijk
    q = vector([1.0, 0.0, 0.0, 0.0])

    q[0] = 0.5 * sqrt(1.0 + m[0][0] - m[1][1] - m[2][2])
    q[1] = (1.0 / 4.0 * q[0]) * (m[2][1] - m[1][2])
    q[2] = (1.0 / 4.0 * q[0]) * (m[0][2] - m[2][0])
    q[3] = (1.0 / 4.0 * q[0]) * (m[1][0] - m[0][1])

    return q

show( QuatFromMatrix( Rprod ) )
(0.707106781186548, 0.000000000000000, 0.000000000000000, 0.000000000000000)\displaystyle \left(0.707106781186548,\,0.000000000000000,\,0.000000000000000,\,0.000000000000000\right)

show( (Rx * Ry * Rz).subs( x == 0, y == pi, z == 0 ) )
show( (Rz * Ry * Rx).subs( x == 0, y == pi, z == 0 ) )
(−10001000−1)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\right)
(−10001000−1)\displaystyle \left(\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array}\right)