We provide numerical bounds on the Crouzeix ratiofor KLS matrices $A$ which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials $p$ of the spectral norm of $p(A)$ dividedby the maximum absolute value of $p$ on the numerical range of $A$.Our bounds confirm the conjecture that this ratiois less than or equal to $2$. We also give a precise description of these numerical ranges.

翻译：我们针对一类在其数值范围边界上包含线段的KLS矩阵$A$，给出了Crouzeix比值的数值界。Crouzeix比值定义为所有多项式$p$的谱范数$\|p(A)\|$与$p$在$A$数值范围内最大绝对值之比的上确界。我们的数值界证实了该比值不超过$2$的猜想，并精确描述了这些数值范围的结构。