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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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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.

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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?

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