Consider the matrix \(A = \left[ \begin{array} { l l } { a } & { b } \\ { c } & { d } \end{array} \right] , a , b , c , d \in \mathbb { R }\) , det \(A = 1\) . It transforms the basis vectors \({ \vec { e } } _ { 1 } = \left[ { \begin{array} { l } { 1 } \\ { 0 } \end{array} } \right] , { \vec { e } } _ { 2 } = \left[ { \begin{array} { l } { 0 } \\ { 1 } \end{array} } \right]\) into \({ \vec { g } } _ { 1 } = { \Big [ } { \frac { a } { c } } { \Big ] } , { \vec { g } } _ { 2 } = { \Big [ } { \frac { b } { d } } { \Big ] }\) . One can obtain a useful decomposition of A by making an inverse transformation in three steps.
Construct the rotation matrix \(R ^ { - 1 }\) that sends \(\vec { g } _ { 1 }\) to \(R ^ { - 1 } ( \vec { g } _ { 1 } ) \lVert \vec { e } _ { 1 }\) . How does it act on \(\vec { g } _ { 2 } ?\)
Construct the diagonal matrix \(P ^ { - 1 }\) such that det \(P ^ { - 1 } = 1\) and \(P ^ { - 1 } ( R ^ { - 1 } ( \vec { g } _ { 1 } ) ) =\) \(\vec { e } _ { 1 }\) . Show that the components of \(P ^ { - 1 } ( R ^ { - 1 } ( \vec { g } _ { 2 } ) )\) are \({ \binom { x } { 1 } } , x \in \mathbb { R }\)
Apply a shear transformation \(T ^ { - 1 }\) that leaves \(\vec { e } _ { 1 }\) invariant and sends \(P ^ { - 1 } ( R ^ { - 1 } ( \vec { g } _ { 2 } ) )\) to \(\vec { e } _ { 2 }\) Then, we have \(T ^ { - 1 } P ^ { - 1 } R ^ { - 1 } A = E\) ,or \(A = R P T\)
Show that this decomposition is unique.