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kmcrane

Really, it's the ratio of signed volumes. For instance, the identity $I$ and minus the identity $-I$ have opposite determinants. What does that mean geometrically? In other words, what's the difference between hitting a figure with $I$ versus $-I$?

pavelkang

I think $I$ means rotation by $0$ degrees and $-I$ means reflection around myself

kmcrane

Ok, actually I should be more careful here because the slide is in 2D and my comment is in 3D: (I) and (-I) have opposite determinants in 3D (1 and -1, respectively), but the same determinant in 2D (both 1). How can this be the case?

If you can make sense of this example, you'll be well on your way to understanding determinants, rotations, and reflections.

pavelkang

In 2D, multiplying a shape by $-I$ makes $(x, y)$ $(-x, -y)$, which is rotation by $\pi$. So it does not change the orientation. In 3D, determinant can be thought of as the volume of a parallelepiped, and negating the three edges of it negates its volume. Or I think we can use $a x b$ to find the orientation in 2D. If we negate all coordinates, $a x b$ does not change its sign. In 3D, we can use $a \dot (b x c)$ to find the orientation, which is another way to write the determinant of a matrix with rows a, b, c. Well I think I put down a lot of thoughts without directly answering your question. So what is an intuitive way to think about this?

kmcrane

@pavelkang: I think those are all good thoughts. Another way to do it is to say: in n dimensions, minus the identity matrix $-I$ can be expressed as the product of n elementary reflections about each of the axes. E.g.,

$$\left[ \begin{array}{rrr} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1 \end{array} \right] = \left[ \begin{array}{rrr} -1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right]\left[ \begin{array}{rrr} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 1 \end{array} \right]\left[ \begin{array}{rrr} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1 \end{array} \right].$$

Since each reflection reverses the orientation (and reversing the orientation twice preserves it), (-I) is orientation-reversing in an odd number of dimensions and orientation-preserving in an even number of dimensions.

kmcrane

Well, my matrices didn't format correctly... but hopefully you get the idea. :-)