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 \delta_{ij} 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 × n shift matrix S becomes the zero 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.)

Properties

Let L and U be the n × n lower and upper shift matrices, respectively. The following properties hold for both U and L. Let us therefore only list the properties for U: The following properties show how U and L are related: If N is any nilpotent matrix, then N is similar to a block diagonal matrix of the form where each of the blocks S1, S2, ..., Sr is a shift matrix (possibly of different sizes).

Examples

Then, Clearly there are many possible permutations. For example, is equal to the matrix A shifted up and left along the main diagonal.