Suppose we have $A_i$, $i=1\ldots n$, $n\times n$ complex matrices linearly independent. It may be conjectured that there exist $(a_1,\ldots,a_n) \in \mathbb{C}^n$ not all zero such that $\sum_{i=1}^na_iA_i$ has a one eigenvalue of multiplicity $n$. I didn't try many things as the linearly independent condition is necessary to assume (hard to use). The question is it provable or do any related facts exist. Thank you.

Suppose we have $A_i$, $i=1\ldots n$, $n\times n$ complex matrices linearly independent. It may be conjectured that there exist $(a_1,\ldots,a_n) \in \mathbb{C}^n$ not all zero such that $\sum_{i=1}^na_iA_i$ has a one eigenvalue of (algebraic) multiplicity $n$. I didn't try many things as the linearly independent condition is necessary to assume (hard to use). The question is it provable or do any related facts exist. Thank you.