How strong does a matrix distort angles? How strong does it distort lengths anisotrolicly? let there be given a square matrix $M \in \mathbb R^{N\times N}$. I would like to have some kind of measure in how far it


*

*Distorts angles between vectors

*It stretches and squeezes discriminating directions.


While I am fine with $M = S \cdot Q$, with $S$ being a positive multiple of the identity and $Q$ being an orthogonal matrix, I would like to measure in how far $M$ diverts from this form. In more visual terms, I would like to measure by a numerical expression to what extent a set is non-similar to the image of the respective set.
What I have in mind is a numerical measure, just like, e.g, the determinant of $M$ measures the volume of a cube under the transform by $M$. Can you help me?

Consider the singular value decomposition of the matrix. 
Look at the singular values. These tell you, how the unit sphere is 
stretched or squeezed by the matrix along the directions corresponding to the
singular vectors.
