Reduced dimension and multiple eigenvalues of a diagonizable \(4\times 4\) matrix (Q1588721)

Let \(a_1,\dots, a_n\) be real \(m\times m\) matrices and let \(d\) be the dimension of the vector space spanned by them and the unit matrix. Let \(r\) be an integer such that \(2\leq r\leq m\). In the case where the \(a_i\) are symmetric, \textit{S. Friedland} and \textit{R. Loewy} [Pac. J. Math. 62, 389-399 (1976; Zbl 0355.15003)] proved that, if \(d\) is sufficiently large, there is a linear combination \(\sum\xi_ia_i\neq 0\), whose largest eigenvalue has multiplicity at least \(r\). The authors' aim is to prove the same result under the weaker hypothesis that every linear combination of the \(a_i\) is similar to a real diagonal matrix. They do this in the present paper in the special case \(m=4\), \(r=3\) and state that ``the general case is in preparation''.