Geodesic
Expansion of vectors in powers of a~matrix
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 199-218

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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. 
@article{ZNSL_2008_355_a8,
     author = {I. E. Maksimenko and E. L. Rabkin},
     title = {Expansion of vectors in powers of a~matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {199--218},
     publisher = {mathdoc},
     volume = {355},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a8/}
}
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I. E. Maksimenko; E. L. Rabkin. Expansion of vectors in powers of a~matrix. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 36, Tome 355 (2008), pp. 199-218. http://geodesic.mathdoc.fr/item/ZNSL_2008_355_a8/