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 ${{𝒞}}_n$ of dimension $n$ under the existence of the moment generating function of the random entries is $\kappa ({{𝒞}}_n)=\text{O}(\frac{1}{𝜀}{n}^{\rho +1/2}{(logn)}^{1/2})$ with probability $1-\text{O}(({𝜀}^2+𝜀)n^{-2\rho }+n^{-1/2+\text{o}(1)})$ for any $𝜀>0$, $\rho \in (0,1/4)$. Moreover, if the random entries only have the second moment, the condition number satisfies $\kappa ({{𝒞}}_n)=\text{O}(\frac{1}{𝜀}{n}^{\rho +1/2}logn)$ with probability $1-\text{O}(({𝜀}^2+𝜀)n^{-2\rho }+{(logn)}^{-1/2})$. Also, we analyze the condition number of a random symmetric circulant matrix ${{𝒞}}_n^{\mathrm{\text{sym}}}$. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix ${{𝒯}}_n$ we establish $\kappa ({{𝒯}}_n)\le \kappa ({{𝒞}}_{2n}){({\sigma }_{min}(C_{2n}){\sigma }_{min}({{𝒮}}_n))}^{-1}$, where ${\sigma }_{min}(A)$ is the minimum singular value of the matrix $A$. The matrix $C_{2n}$ is a random circulant matrix and ${{𝒮}}_n:=F_{2,n}^{\ast }{D}_{1,n}^{-1}{F}_{2,n}+F_{4,n}^{\ast }{D}_2^{-1}{F}_{4,n}$, where $F_{2,n},F_{4,n}$ are deterministic matrices, $F^{\ast }$ indicates the conjugate transpose of $F$ and $D_{1,n},D_{2,n}$ are random diagonal matrices. From random experiments, we conjecture that ${{𝒮}}_n$ is well-conditioned if the moment generating function of the random entries of ${{𝒞}}_{2n}$ exists.

在本文中，我们研究随机 Toeplitz 矩阵的条件数。由于托普利兹矩阵是一个对角常数矩阵，它的行或列不能随机独立。这种情况不允许我们在随机矩阵的所有条目随机独立时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。循环矩阵实际上是托普利茨矩阵的一种特殊情况，但具有更具体的结构，在这种情况下，可以获得其特征值和奇异值的明确公式。在我们的结果中，${{𝒞}}_n$ 维度的 $n$ 在随机条目的矩生成函数存在的情况下 $\kappa ({{𝒞}}_n)=\text{哦}(\frac{1}{𝜀}{n}^{\rho +1/2}{(日志n)}^{1/2})$ 有概率 $1-\text{哦}(({𝜀}^2+𝜀)n^{-2\rho }+n^{-1/2+\text{○}(1)})$ 对于任何 $𝜀>0$, $\rho \in (0,1/4)$. 而且，如果随机条目只有二阶矩，则条件数满足$\kappa ({{𝒞}}_n)=\text{哦}(\frac{1}{𝜀}{n}^{\rho +1/2}日志n)$ 有概率 $1-\text{哦}(({𝜀}^2+𝜀)n^{-2\rho }+{(日志n)}^{-1/2})$. 另外，我们分析了一个随机对称循环矩阵的条件数${{𝒞}}_n^{\mathrm{\text{符号}}}$. 对于随机（非对称或对称）托普利兹矩阵的条件数${{𝒯}}_n$ 我们建立 $\kappa ({{𝒯}}_n)\le \kappa ({{𝒞}}_{2n}){({\sigma }_{分钟}(C_{2n}){\sigma }_{分钟}({{𝒮}}_n))}^{-1}$， 在哪里 ${\sigma }_{分钟}(\mathrm{一种})$ 是矩阵的最小奇异值 $\mathrm{一种}$. 矩阵$C_{2n}$ 是一个随机循环矩阵，并且 ${{𝒮}}_n：=F_{2,n}^{＊}{D}_{1,n}^{-1}{F}_{2,n}+F_{4,n}^{＊}{D}_2^{-1}{F}_{4,n}$， 在哪里 $F_{2,n},F_{4,n}$ 是确定性矩阵， $F^{＊}$ 表示共轭转置 $F$ 和 $D_{1,n},D_{2,n}$是随机对角矩阵。根据随机实验，我们推测${{𝒮}}_n$ 如果随机条目的矩生成函数是良好条件的 ${{𝒞}}_{2n}$ 存在。