In this paper, we restate a necessary and sufficient condition for all positive integer powers of a Toeplitz matrix to be still Toeplitz matrices which was ever studied by T. N. E. Greville (1983) and Tamir Shalom (1987), and our new proof is? clear and self-contained. We also derive some identities involving the entries of the Toeplitz matrix and those of its positive integer powers in case that the powers are still Toeplitz matrices.
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