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3D Decompositions

Let us consider an arbitrary (not zero) matrix

\begin{equation} \label{Eq2D.6} \mathbf{A} = \begin{bmatrix} \mathbf{u} & \mathbf{v} & \mathbf{w} \end{bmatrix} , \quad\mbox{where} \quad \mathbf{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} , \quad \mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} , \quad \mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} . \end{equation}
Using spherical coordinates, we rewtite the matrix A as
\[ \mathbf{A} = \begin{bmatrix} \cos\theta_u \sin\phi_u & \cos\theta_v \sin\phi_v & \cos\theta_w \sin\phi_w \\ \sin\theta_u \sin\phi_u & \sin\theta_v \sin\phi_v & \sin\theta_w \sin\phi_w \\ \cos\phi_u & \cos\phi_v & \cos\phi_w \end{bmatrix} \cdot \begin{bmatrix} \| \mathbf{u} \| & 0 & 0 \\ 0 & \| \mathbf{v} \| & 0 \\ 0&0& \| \mathbf{w} \| \end{bmatrix}, \]
where
\[ \| \mathbf{u} \| = +\sqrt{u_1^2 + u_2^2 + u_3^2} , \quad \| \mathbf{v} \| = +\sqrt{v_1^2 + v_2^2 + v_3^2} , \quad \| \mathbf{w} \| = +\sqrt{w_1^2 + w_2^2 + w_3^2} , \]
and
\[ 0 \le \phi = \mbox{arcos} \frac{z}{\| \cdot \|} < \pi , \qquad 0 \le \theta = \mbox{sign}(y)\,\mbox{arccos} \frac{x}{\sqrt{x^2 + y^2}} < 2\pi . \]
   
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