In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling technique, we break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. A circulant matrix is, in fact, a particular case of a Toeplitz matrix, but with a more specific structure, where it is possible to obtain explicit formulas for its eigenvalues and also for its singular values. Among our results, we show the condition number of a non-symmetric random circulant matrix 𝒞n of dimension n under the existence of the moment generating function of the random entries is κ(𝒞n) = O(1 𝜀nρ+1/2(log n)1/2) with probability 1 −O((𝜀2 + 𝜀)n−2ρ + n−1/2+o(1)) for any 𝜀 > 0, ρ ∈ (0, 1/4). Moreover, if the random entries only have the second moment, the condition number satisfies κ(𝒞n) = O(1 𝜀nρ+1/2log n) with probability 1 −O((𝜀2 + 𝜀)n−2ρ + (log n)−1/2). Also, we analyze the condition number of a random symmetric circulant matrix 𝒞nsym. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix 𝒯n we establish κ(𝒯n) ≤ κ(𝒞2n)(σmin(C2n)σmin(𝒮n))−1, where σmin(A) is the minimum singular value of the matrix A. The matrix C2n is a random circulant matrix and 𝒮n := F2,n∗D 1,n−1F 2,n + F4,n∗D 2−1F 4,n, where F2,n,F4,n are deterministic matrices, F∗ indicates the conjugate transpose of F and D1,n,D2,n are random diagonal matrices. From random experiments, we conjecture that 𝒮n is well-conditioned if the moment generating function of the random entries of 𝒞2n exists.

中文翻译：

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随机 Toeplitz 矩阵：高度随机依赖下的条件数

在本文中，我们研究了随机 Toeplitz 矩阵的条件数。由于 Toeplitz 矩阵是对角常数矩阵，它的行或列不能是随机独立的。当随机矩阵的所有元素都随机独立时，这种情况不允许我们使用经典策略来分析其最小奇异值。使用循环嵌入作为解耦技术，我们打破了 Toeplitz 矩阵结构的随机依赖性，并减少了分析随机循环矩阵的极端奇异值的问题。实际上，循环矩阵是 Toeplitz 矩阵的一种特殊情况，但具有更具体的结构，可以为其特征值和奇异值获得显式公式。在我们的结果中，𝒞n维度的n在随机条目的矩生成函数存在下是κ(𝒞n) = ○(1 𝜀nρ+1/2(日志 n)1/2)有概率1 -○((𝜀2 + 𝜀)n-2ρ + n-1/2+○(1))对于任何𝜀 > 0,ρ ∈ (0, 1/4). 此外，如果随机条目只有二阶矩，则条件数满足κ(𝒞n) = ○(1 𝜀nρ+1/2日志 n)有概率1 -○((𝜀2 + 𝜀)n-2ρ + (日志 n)-1/2). 此外，我们分析了随机对称循环矩阵的条件数𝒞n符号. 对于随机（非对称或对称）Toeplitz 矩阵的条件数𝒯n我们建立κ(𝒯n) ≤ κ(𝒞2n)(σ分钟(C2n)σ分钟(𝒮n))-1， 在哪里σ分钟(一个)是矩阵的最小奇异值一个. 矩阵C2n是一个随机循环矩阵并且𝒮n ：= F2,n*D 1,n-1F 2,n + F4,n*D 2-1F 4,n， 在哪里F2,n,F4,n是确定性矩阵，F*表示共轭转置F和D1,n,D2,n是随机对角矩阵。从随机实验中，我们推测𝒮n如果随机条目的矩生成函数是良条件的𝒞2n存在。