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 \( \delta _{ij} \) 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 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:

det(U) = 0
trace(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:

LT = U; UT = L
The null spaces of U and L are

\( {\displaystyle N(U)=\operatorname {span} \left\{(1,0,\ldots ,0)^{\mathsf {T}}\right\},} \)
\( {\displaystyle N(L)=\operatorname {span} \left\{(0,\ldots ,0,1)^{\mathsf {T}}\right\}.} \)

The spectrum of U and L is \( \{0\} \). The algebraic multiplicity of 0 is n, and its geometric multiplicity is 1. From the expressions for the null spaces, it follows that (up to a scaling) the only eigenvector for U is \( {\displaystyle (1,0,\ldots ,0)^{\mathsf {T}}} \), and the only eigenvector for L is \( {\displaystyle (0,\ldots ,0,1)^{\mathsf {T}}}. \)
For LU and UL we have
\( {\displaystyle UL=I-\operatorname {diag} (0,\ldots ,0,1),} \)
\( {\displaystyle LU=I-\operatorname {diag} (1,0,\ldots ,0).} \) These matrices are both idempotent, symmetric, and have the same rank as U and L
Ln−aUn−a + LaUa = Un−aLn−a + UaLa = I (the identity matrix), for any integer a between 0 and n inclusive.

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, ..., Sr is a shift matrix (possibly of different sizes).[2][3]
Examples

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

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.

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