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 formula_2 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.)
Properties.
Let "L" and "U" be the "n" by "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 "S"1, "S"2, ..., "S""r" is a shift matrix (possibly of different sizes).
Examples.
Then,
Clearly there are many possible permutations. For example, formula_8 is equal to the matrix "A" shifted up and left along the main diagonal.