Classification of matrices with two distinct eigenvalues under equitable partitions

Classify all matrices M in C^{n×n} having exactly two distinct eigenvalues with multiplicities {α^{[i]}, β^{[n−i]}} for 2 ≤ i ≤ ⌊n/2⌋, under the equitable partition π = { {1,2,…,i}, {i+1,…,n} }, in the sense of determining when such matrices arise in the framework developed for Theorem \ref{thm:n-by-n-two-eigs}.

Background

Section 3 develops constructive characterizations of matrices whose equitable quotient contains two distinct eigenvalues, including a complete treatment of the special case where one eigenvalue has multiplicity n−1 (Theorem \ref{thm:n-by-n-two-eigs}).

Immediately thereafter, the text notes that extending this to a full classification for general multiplicity splits {α^{[i]}, β^{[n−i]}} (and further to three or more distinct eigenvalues) remains unresolved, highlighting the challenge of handling broader multiplicity patterns and partitions.