I am interested to know if there exist a notion of $k$-minors of a real square matrix, for non-integer positive values of $k$

One approach I thought of was to use the fact that the $k$-minors are (roughly) the derivative of the $k+1$ minors. (i.e. the derivative of the $k+1$ exterior power map $A \to \bigwedge^{k+1} A$ encodes $\bigwedge^{k} A$).

This might be generalized using fractional derivatives. However, when I tried some calculation it got nowhere; since we are talking about polynomials, the natural fractional derivative gives infinity when evaluated at zero).

What I have in mind is an assignment $M_{\alpha}: \text{End}(\mathbb{R}^n) \to \mathbb{R}^{?}$, which has (some of) the following properties:

  1. It scales as expected, i.e. $M_{\alpha} (cA)=c^{\alpha}M_{\alpha}(A)$,

  2. If $\text{rank} (A) < \alpha$ then $M_{\alpha} (A)=0$. (maybe this should be replaced by $\text{rank} (A)<[\alpha]$ where $[\alpha]$ is the integer part of $\alpha$).

  3. $\det A$ can be expressed via $M_{\alpha}(A),M_{\dim A-\alpha}(A),$

Again, I am not sure exactly what I am looking for. Has such a notion been studied somewhere?

  • $\begingroup$ I suppose that your condition 1. should be read as $M_\alpha(cA)=c^\alpha M_\alpha (A)$ for all $A$ ? $\endgroup$ – GreginGre Nov 22 '18 at 12:59
  • $\begingroup$ Yes, thanks; this was a typo. fixed now. $\endgroup$ – Asaf Shachar Nov 22 '18 at 13:13

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