# Obtain angles of rotation from singular vectors U and V from SVD

My question is a continuation to a previous question. I have a matrix M which is a function of theta1, theta2, phi1 and phi2. Now, I want to get back these angles using the singular vectors (U and V) of an SVD ($$U.W.V^{T}$$). I want to start by taking a random matrix M and try to extract these angles by manipulating matrices U and V. I however took it a bit different way. I tried to come up with a combination of rotations T1 and T2 that gives a similar matrix as U and V but not exactly the same. So, I cannot equate them as I had initially thought. Here's my code:

M={{161.294, 11.3761, 60.498}, {-10.7667, -7.11777, 30.0436}, {-64.1061,
29.8182, -15.9093}};

{U, W, V} = SingularValueDecomposition[M];
U
V

Uy[theta] = {{Cos[theta]],0,Sin[theta]},{0,1,0},{-Sin[theta],0,Cos[theta]}};
Uz[phi] = {{Cos[phi],-Sin[phi],0},{Sin[phi],Cos[phi],0},{0,0,1}};
T1 = Uy[theta1]
T2 = Uy[theta2].Uz[phi1].Uz[phi2]


Any help shall be highly appreciated.

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• Avoid using MatrixForm (which is really just for pretty printing matrices) when you set variables like T1,T2, as this wraps a MatrixForm head around the expression and it won't work as expected in any future computations you do with these variables. Jan 3 at 22:29