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?

How much $M$ distorts angles between vectors depends not just on $M$, but on the vectors. The extent to which a set is "not similar" to its image depends not just on $M$, but on the set. So I'm not sure there's a coherent answer to your question. 
E.g., if $$M=\pmatrix{1&0\cr0&0\cr}$$ then some big angles get squashed flat while vectors that are already parallel stay parallel; some sets get squashed flat while some sets that are already flat remain unchanged. What would you like the numerical measure of that matrix to be?
