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

${\displaystyle U_{ij}=\delta _{i+1,j},\quad L_{ij}=\delta _{i,j+1},}$

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta symbol.

For example, the 5 × 5 shift matrices are

${\displaystyle U_{5}={\begin{pmatrix}0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&0&0&0\end{pmatrix}}\quad L_{5}={\begin{pmatrix}0&0&0&0&0\\1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\end{pmatrix}}.}$

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.¹

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:

• det(U) = 0
• tr(U) = 0
• rank(U) = n − 1
• The characteristic polynomials of U is

  ${\displaystyle p_{U}(\lambda )=(-1)^{n}\lambda ^{n}.}$

• Uⁿ = 0. This follows from the previous property by the Cayley–Hamilton theorem.
• The permanent of U is 0.

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

${\displaystyle {\begin{pmatrix}S_{1}&0&\ldots &0\\0&S_{2}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &S_{r}\end{pmatrix}}}$

where each of the blocks S₁, S₂, ..., S_r is a shift matrix (possibly of different sizes).²³

${\displaystyle S={\begin{pmatrix}0&0&0&0&0\\1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\end{pmatrix}};\quad A={\begin{pmatrix}1&1&1&1&1\\1&2&2&2&1\\1&2&3&2&1\\1&2&2&2&1\\1&1&1&1&1\end{pmatrix}}.}$

Then,

${\displaystyle SA={\begin{pmatrix}0&0&0&0&0\\1&1&1&1&1\\1&2&2&2&1\\1&2&3&2&1\\1&2&2&2&1\end{pmatrix}};\quad AS={\begin{pmatrix}1&1&1&1&0\\2&2&2&1&0\\2&3&2&1&0\\2&2&2&1&0\\1&1&1&1&0\end{pmatrix}}.}$

Clearly there are many possible permutations. For example, ${\displaystyle S^{\mathsf {T}}AS}$ is equal to the matrix A shifted up and left along the main diagonal.

${\displaystyle S^{\mathsf {T}}AS={\begin{pmatrix}2&2&2&1&0\\2&3&2&1&0\\2&2&2&1&0\\1&1&1&1&0\\0&0&0&0&0\end{pmatrix}}.}$