Expansion of vectors by powers of a matrix

In this paper, we investigate the problem of expansion of any d-dimensional vector in powers of a dilation matrix M, where a dilation matrix is an integer matrix of size d × d with all modules of its eigenvalues more than one. We consider this expansion as a multidimensional system of numeration, where we take the matrix as the base of the system of numeration and a special set of vectors as the set of digits. We give a constructive method of expansion of integer vectors in powers of a dilation matrix and prove the existence of expansion for any real vector. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 199–218.