General properties of eigenvalues of \(A+\tau uv^*\) as functions of \(\tau \in {\mathbb {C} }\) or \(\tau \in {\mathbb {R} }\) or \(\tau ={{\,\mathrm{{e}}\,}}^{{{\,\mathrm{{i}}\,}}\theta }\) on the unit circle are considered. In particular, the problem of existence of global analytic formulas for eigenvalues is addressed. Furthermore, the limits of eigenvalues with \(\tau \rightarrow \infty \) are discussed in detail. The following classes of matrices are considered: complex (without additional structure), real (without additional structure), complex H-selfadjoint and real J-Hamiltonian.

\（A + \ tau uv ^ * \）的特征值的一般属性是\（\ tau \ in {\ mathbb {C}} \）或\（\ tau \ in {\ mathbb {R}} \}的函数，或者考虑单位圆上的\（\ tau = {{\，\ mathrm {{e}} \，}} ^ {{{\，\ mathrm {{i}} \，}} \ theta} \）。特别地，解决了特征值的全局解析公式的存在的问题。此外，详细讨论了\（\ tau \ rightarrow \ infty \）的特征值的限制。考虑以下几类矩阵：复数（无附加结构），实数（无附加结构），复数H-自伴和实数J-哈密​​顿量。