# 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

1. Distorts angles between vectors
2. 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?

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(+1) One function of the singular values that is invariant under scalar multiplication of $Q$ is the absolute value of the ratio of the smallest and largest singular values. This would reply to #2. To address #1, one could minimize $<v,Mw>$ over all unit vectors for which $<v,w>=0$ (which is closely related to a standard geodetic measurement used to assess projections of an ellipsoid). –  whuber Nov 9 '11 at 5:01
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?