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
Uij=\deltai+1,j, Lij=\deltai,j+1,
where
\deltaij
For example, the 5 × 5 shift matrices are
U5=\begin{pmatrix} 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{pmatrix} L5=\begin{pmatrix} 0&0&0&0&0\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0 \end{pmatrix}.
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.)
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:
pU(λ)=(-1)nλn.
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
\begin{pmatrix}S1&0&\ldots&0\ 0&S2&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&Sr\end{pmatrix}
where each of the blocks S1, S2, ..., Sr is a shift matrix (possibly of different sizes).
S=\begin{pmatrix} 0&0&0&0&0\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0 \end{pmatrix}; A=\begin{pmatrix} 1&1&1&1&1\\ 1&2&2&2&1\\ 1&2&3&2&1\\ 1&2&2&2&1\\ 1&1&1&1&1 \end{pmatrix}.
Then,
SA=\begin{pmatrix} 0&0&0&0&0\\ 1&1&1&1&1\\ 1&2&2&2&1\\ 1&2&3&2&1\\ 1&2&2&2&1 \end{pmatrix}; AS=\begin{pmatrix} 1&1&1&1&0\\ 2&2&2&1&0\\ 2&3&2&1&0\\ 2&2&2&1&0\\ 1&1&1&1&0 \end{pmatrix}.
Clearly there are many possible permutations. For example,
STAS
STAS=\begin{pmatrix} 2&2&2&1&0\\ 2&3&2&1&0\\ 2&2&2&1&0\\ 1&1&1&1&0\\ 0&0&0&0&0 \end{pmatrix}.