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Polar decomposition in 2D

By the polar decomposition theorem, every nonsingular real square matrix A can be factored uniquely into the product of an orthogonal and a positive definite real matrix:
\begin{equation} \label{EqPolar.1} \mathbf{M} = {\bf V}\,{\bf P}, \qquad \mathbf{V}^T {\bf V} = {\bf I} \quad\& \quad \mathbf{P} = \mathbf{P}^T > 0 , \end{equation}
where VT and PT denote the transposes of V and P.
Theorem 1: Every nonsingular two-dimensional matrix M ∈ ℝ2×2 has the polar decomposition (1) where \[ \mathbf{P} = \left\vert \mathbf{M} + \vert \det\mathbf{M} \vert \left( \mathbf{M}^{\mathrm T} \right)^{-1} \right\vert^{-1/2} \left( \mathbf{M}^{\mathrm T} \mathbf{M} + \vert \det \mathbf{M} \vert \mathbf{I} \right) \] and \[ \mathbf{V} = \left\vert \det \left( \mathbf{M} + \left\vert \det\mathbf{M} \right\vert \left( \mathbf{M}^{\mathrm T} \right)^{-1} \right) \right\vert^{-1/2} \left( \mathbf{M} + \vert \det\mathbf{M} \vert \left( \mathbf{M}^{\mathrm T} \right)^{-1} \right) . \]
Instead of giving the lengthy and rather tedious arithmetic sug- gested by the constructive proof for the polar decomposition theorem here, we need only verify that V and P as given above are in fact the polar factors of any nonsingular \[ \mathbf{M} = \begin{bmatrix} a & b \\ c& d \end{bmatrix} \in \mathbb{R}^{2\times 2} . \] Clearly P is positive definite as the sum of positive definite matrices, and \[ \mathbf{V}^{\mathrm T} \mathbf{V} = \left\vert \mathbf{M} + \vert \det\mathbf{M} \vert \left( \mathbf{M}^{\mathrm T} \right)^{-1} \right\vert^{-1} . \] \[ \times \left[ \mathbf{M}^{\mathrm T} \mathbf{M} + 2 \right] . \]