The characteristic values of the p-th compound matrix

Question

If $\mathfrak{A_p}$ is the p-th compound matrix of A, May you please explain what's going on here?

I can follow until $$\lim_{m \rightarrow \infty} \lambda_{k_1}^{(m)} \lambda_{k_2}^{(m)} \dots \lambda_{k_p}^{(m)} = \lambda_{k_1} \lambda_{k_2} \dots \lambda_{k_p}$$ But after that, I don't understand what the author mean.

Answer 1

Let $V$ be a complex $n$-dimensional vector space. What I get out of this in modern language is that:

• the linear transformations $V\to V$ with $n$ distinct eigenvalues are dense in the space of all linear transformations,
• the map taking a linear transformation $A:V\to V$ to its multiset of eigenvalues is continuous (in Hausdorff distance, say),
• the map taking a linear transformation $A:V\to V$ to its $p$th exterior power $\bigwedge^p A:\bigwedge^p V \to \bigwedge^p V$ is also continuous (the author uses $\mathfrak{A}_p$ and "compound matrix" for this latter concept), and
• if a linear transformation $A:V\to V$ has $n$ distinct eigenvalues, then the eigenvalues of $\bigwedge^p A:\bigwedge^p V\to \bigwedge^p V$ are given by all products of $p$-element subsets of the set of eigenvalues of $A$.

Consequently, the author concludes that for any linear transformation $A$ (whose eigenvalues need not be distinct), the eigenvalues of $\bigwedge^p A:\bigwedge^p V\to \bigwedge^p V$ are given by all products of $p$-element sub-multisets of the multiset of eigenvalues of $A$ (accounting properly for multiplicity). To draw this conclusion, the author uses density and continuity to reduce to the case of $n$ distinct eigenvalues for which the result has already been proven.