Expansion of vectors in powers of a matrix

In this paper, we investigate the problem of expansion of any $d$-dimensional vector in powers of a dilation matrix $M$. (A dilation matrix is an integral matrix of size $d\times d$ with all eigenvalues greater than 1 in modulus.) This expansion can be viewed as a multidimensional system of numeration with the matrix as the base and a special set of vectors as the set of digits. We give a constructive method of expanding an integral vector in powers of a dilation matrix and prove the existence of an expansion for any real vector. Bibl. – 4 titles.