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 shift matrices are nilpotent; an n by n shift matrix S becomes the null matrix when raised to the power of its dimension n.

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:

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

The following properties show how U and L are related:

${\displaystyle N(U)=\operatorname {span} \{(1,0,\ldots ,0)^{T}\},}$
${\displaystyle N(L)=\operatorname {span} \{(0,\ldots ,0,1)^{T}\}.}$
• The spectrum of U and L is ${\displaystyle \{0\}}$. 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 ${\displaystyle (1,0,\ldots ,0)^{T}}$, and the only eigenvector for L is ${\displaystyle (0,\ldots ,0,1)^{T}}$.
• 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 S1S2, ..., 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, ${\displaystyle S^{T}AS}$ is equal to the matrix A shifted up and left along the main diagonal.

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