The solution of the Loewy–Radwan conjecture (Q6622102)

The aim of this paper is to prove the Loewy-Radwan conjecture, see [\textit{R. Loewy} and \textit{N. Radwan}, Electron. J. Linear Algebra 3, 142--152 (1998; Zbl 0913.15001)]. This conjecture is the following: Let \(n\) and \(k<n\) be positive integers and let \(V\) be a linear subspace of \(M_n(\mathbb{C})\) with the property that each member of \(V\) has at most \(k\) distinct eigenvalues. Then\N\[\N\dim V \leq \binom{n}{2} + \binom{k}{2} + 1.\N\]\NMoreover, if the equality holds and \(k \geq 3\), then there exists \(p \in \{0, 1, \ldots , n-k + 1\}\) such that \(V\) is simultaneously similar to the space of all matrices of the form\N\[\N\left( \begin{array}{ccc} A & B & C \\\N0 & D & E \\\N0 & 0 & F \end{array} \right)\N\]\Nwhere \(B \in M_{p \times (k-1)}(\mathbb{C})\), \(D \in M_{k-1}(\mathbb{C})\) and \(E \in M_{(k-1) \times (n-k-p+1)}(\mathbb{C})\) are arbitrary and\N\[\N\left( \begin{array}{cc} A & C \\\N0 & F \end{array} \right)\N\]\Nis an arbitrary upper triangular matrix with equal diagonal entries.