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?