Sparse matrices | Matrices

Shift matrix

In mathematics, a shift matrix is a binary matrix with ones only on the superdiagonal or subdiagonal, and zeroes elsewhere. A shift matrix U with ones on the superdiagonal is an upper shift matrix. The alternative subdiagonal matrix L is unsurprisingly known as a lower shift matrix. The (i,j):th component of U and L are where is the Kronecker delta symbol. For example, the 5×5 shift matrices are Clearly, the transpose of a lower shift matrix is an upper shift matrix and vice versa. As a linear transformation, a lower shift matrix shifts the components of a column vector one position down, with a zero appearing in the first position. An upper shift matrix shifts the components of a column vector one position up, with a zero appearing in the last position. Premultiplying a matrix A by a lower shift matrix results in the elements of A being shifted downward by one position, with zeroes appearing in the top row. Postmultiplication by a lower shift matrix results in a shift left.Similar operations involving an upper shift matrix result in the opposite shift. Clearly all finite-dimensional shift matrices are nilpotent; an n by n shift matrix S becomes the null matrix when raised to the power of its dimension n. Shift matrices act on shift spaces. The infinite-dimensional shift matrices are particularly important for the study of ergodic systems. Important examples of infinite-dimensional shifts are the Bernoulli shift, which acts as a shift on Cantor space, and the Gauss map, which acts as a shift on the space of continued fractions (that is, on Baire space.) (Wikipedia).

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Related pages

Cantor space | Permanent (mathematics) | Characteristic polynomial | Trace (linear algebra) | Spectral theorem | Continued fraction | Kronecker delta | Identity matrix | Nilpotent | Determinant | Nilpotent matrix | Cayley–Hamilton theorem | Matrix similarity | Gauss–Kuzmin–Wirsing operator | Mathematics | Subshift of finite type | Baire space (set theory) | Shift space | Rank (linear algebra)