Why do odd dimensions and even dimensions behave differently?

It is well known that odd and even dimensions work differently.

Is there some unifying reason for these differences? Why is dimensional parity so important?

(If you know of any more interesting differences, please leave a comment and I will add them to the list.)

Answer

For real vector spaces, i.e. $\mathbb{R}^N$, one important difference between odd and even $N$ is that every real polynomial with odd order has a real zero, while there are real polynomials with even order that only have complex zeros.

Thus, a real invertible $N\times N$ matrix always has a non-zero eigenvector if $N$ is odd (because the characteristic polynomial has order $N$, and by the above has a zero). For even $N$, however, there are invertible real matrices with no non-zero eigenvector.

This means that in even-dimensional spaces, there are invertible linear mappings without an invariant one-dimensional subspace, while in odd-dimensional spaces, invertible linear mappings always have a one-dimensional invariant subspace. If you look only at rotations, you get that rotations in an odd-dimensional space always keep at least a line fixed, while in an even-dimensional space they do not.

That’s a pretty huge geometric difference between odd- and even-dimensional spaces, that should explain a least some of the items in the question.

Attribution
Source : Link , Question Author : Potato , Answer Author : fgp

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