Kippenhahn's Conjecture Revisited

Abstract

In 1951 paper \cite{Ki} Kippenhahn conjectured that if the characteristic polynomial \ $P_A(x_1,x_2,x_3)=\mbox{det}(x_1A_1+x_2A_2-x_3I)$, \ where $A_1$ and $A_2$ are $n\times n$ Hermitian matrices, has a repeated factor in the polynomial ring $\C[x_1,x_2,x_3]$, then the pair $(A_1,A_2)$ is unitary equivalent to a direct sum $(C_1\oplus C_2, \ D_1\oplus D_2)$ where $C_i, D_i\in M_{n_i}(\C) $ for some $1\leq n_i<n, \ n_1+n_2=n, i=1,2$. Kippenhahn verified the conjecture whenever the degree of the minimal polynomial of $x_1A_1 + x_2A_2$ is 1 or 2. In subsequent works \cite{Sh1,Sh2} Shapiro obtained a number of results which supported the conjecture. In particular, she showed that it held if $n \leq 5$. In 1983 Laffey \cite{La} showed that, in general, Kippenhahn's conjecture was not true by constructing a counterexample for $n=8$. Since then additional counterexamples were worked out (see \cite{Wa} for example). Some positive results in this direction including the quantum version of the conjecture can be found in \cite{F1, F2, KVo1, Law}. In this paper we use methods of recently developed local spectral analysis to give some necessary and sufficient conditions for the affirmative answer to Kippenhahn's conjecture in terms of the characteristic polynomials of certain elements of the algebra generated by the matrices in the tuple.