Editing Polar decomposition (section)

==Geometric interpretation==
A real square <math>m\times m</math> matrix <math>A</math> can be interpreted as the [[linear transformation]] of <math>\mathbb{R}^m</math> that takes a column vector <math>x</math> to <math>A x</math>.  Then, in the polar decomposition <math>A = RP</math>, the factor <math>R</math> is an <math>m\times m</math> real orthogonal matrix.  The polar decomposition then can be seen as expressing the linear transformation defined by <math>A</math> into a [[scaling (geometry)|scaling]] of the space <math>\mathbb{R}^m</math> along each eigenvector <math>e_i</math> of <math>P</math> by a scale factor <math>\sigma_i</math> (the action of <math>P</math>), followed by a rotation of <math>\mathbb{R}^m</math> (the action of <math>R</math>).  

Alternatively, the decomposition <math>A=P R</math> expresses the transformation defined by <math>A</math> as a rotation (<math>R</math>) followed by a scaling (<math>P</math>) along certain orthogonal directions.  The scale factors are the same, but the directions are different.