We begin by showing that a n*n matrix can be decomposed into a sum of 'n' circulant matrices with appropriate relaxations. We use Fast-Fourier-Transform (FFT) operations to perform a sparse similarity transformation representing only the dominant circulant components, to evaluate all eigenvalues of dense Toeplitz, block-Toeplitz and other periodic or quasi-periodic matrices, to a reasonable approximation in O(n^2) arithmetic operations. This sparse similarity transformation can be exploited for other evaluations as well.

翻译：我们首先要表明,n*n 矩阵可以分解成“n” 循环矩阵总和,并进行适当的放松。 我们使用快速四轮转换操作来进行稀有的相似性转换,仅代表主要的循环系统组件,对密集的托普利茨、聚块托普利茨和其他定期或半定期基体的所有基因值进行评估,以达到O(n)2算术操作的合理近似值。这种稀有的相似性转换也可以用于其他评估。