We study the spectral properties of flipped Toeplitz matrices of the form $H_n(f)=Y_nT_n(f)$, where $T_n(f)$ is the $n\times n$ Toeplitz matrix generated by the function $f$ and $Y_n$ is the $n\times n$ exchange (or flip) matrix having $1$ on the main anti-diagonal and $0$ elsewhere. In particular, under suitable assumptions on $f$, we establish an alternating sign relationship between the eigenvalues of $H_n(f)$, the eigenvalues of $T_n(f)$, and the quasi-uniform samples of $f$. Moreover, after fine-tuning a few known theorems on Toeplitz matrices, we use them to provide localization results for the eigenvalues of $H_n(f)$. Our study is motivated by the convergence analysis of the minimal residual (MINRES) method for the solution of real non-symmetric Toeplitz linear systems of the form $T_n(f)\mathbf x=\mathbf b$ after pre-multiplication of both sides by $Y_n$, as suggested by Pestana and Wathen.

翻译：本文研究形如 $H_n(f)=Y_nT_n(f)$ 的翻转Toeplitz矩阵的谱性质，其中 $T_n(f)$ 是由函数 $f$ 生成的 $n\times n$ Toeplitz矩阵，$Y_n$ 是 $n\times n$ 交换（翻转）矩阵，其主反对角线上元素为 $1$，其余位置为 $0$。特别地，在 $f$ 满足适当假设的条件下，我们建立了 $H_n(f)$ 的特征值、$T_n(f)$ 的特征值以及 $f$ 的拟均匀样本之间的交替符号关系。此外，通过对Toeplitz矩阵若干已知定理的精细化调整，我们利用这些定理给出了 $H_n(f)$ 特征值的局部化结果。本研究的动机源于Pestana与Wathen提出的对实非对称Toeplitz线性系统 $T_n(f)\mathbf x=\mathbf b$ 进行 $Y_n$ 左乘预处理后，采用最小残差（MINRES）方法求解的收敛性分析。