# A short note on Multilevel Toeplitz Matrices

## TL;DR

This paper extends the known result that one-level Toeplitz matrices are unitarily similar to complex symmetric matrices to multilevel Toeplitz matrices, providing a construction method and exploring related classes.

## Contribution

It introduces a method to construct unitary matrices for multilevel Toeplitz matrices, generalizing previous results and identifying classes of matrices related to Toeplitz structures.

## Key findings

- Multilevel Toeplitz matrices are unitarily similar to complex symmetric matrices.
- A tensor product-based method constructs the necessary unitary matrices.
- A class of matrices similar to multilevel Toeplitz matrices is characterized.

## Abstract

Chien, Liu, Nakazato, and Tam proved that all n by n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper, we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices.