Condition Number of Random Tridiagonal Toeplitz Matrix

Abstract

In this manuscript it is considered the eigenvalues $\lambda_j$ of a random tridiagonal Toeplitz matrix $T$. We study the asymptotic behavior of the joint distribution of $({|{\lambda}|_{\min} ,|{\lambda}|_{\max}})$. From this, we obtain the asymptotic distribution of the condition number when $T$ is symmetric. In the non-symmetric case, we understand well the singularity of the matrix and can give some good estimation of its condition number. It is remarkable that in both these cases, it is only necessary to consider two or three random variables, but this simplicity is apparent since the structure of the tridiagonal Toeplitz matrix provides non-trivial relation between them, which also induce the asymptotic behavior is completely determined by these input random variables. Also, we want to remark that our results are satisfied under mild conditions on the random variables.