# A generalization of matrix minors to non-integer values

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?

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