Algebraic structures related to Toeplitz and Hankel matrices and tensors

This is a first part of the article consisting of two parts. The first part is algebraic and the second part is applications to fast computations and fast prediction. In the first of the paper is shown that although the spaces of Toeplitz and Hankel matrices are of complicated structure from the algebraic point of view, each of them can be decomposed into a sum of two subspaces that are simpler algebraically. In particular, the space of complex square Toeplitz matrices of order $n$ can be represented as a sum of two subspaces intersecting in the space of scalar matrices, each of which is an algebra conjugate to the algebra of complex diagonal matrices of order $n$. Similar results hold in the case of Hankel matrices and also Toeplitz tensors.