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

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

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

For example, the 5 × 5 shift matrices are

$U_5=\left(\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0\end{array}\right)L_5=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right).$

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.)

Properties

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

  $p_U(\lambda )=(-1{)}^n{\lambda }^n.$

• 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: Template:Unordered list

If N is any nilpotent matrix, then N is similar to a block diagonal matrix of the form

$\left(\begin{array}{cccc}{S}_1& 0& \dots & 0\\ 0& S_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & S_r\end{array}\right)$

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

Examples

$S=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right);A=\left(\begin{array}{ccccc}1& 1& 1& 1& 1\\ 1& 2& 2& 2& 1\\ 1& 2& 3& 2& 1\\ 1& 2& 2& 2& 1\\ 1& 1& 1& 1& 1\end{array}\right).$

Then,

$SA=\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 1& 1& 1& 1& 1\\ 1& 2& 2& 2& 1\\ 1& 2& 3& 2& 1\\ 1& 2& 2& 2& 1\end{array}\right);AS=\left(\begin{array}{ccccc}1& 1& 1& 1& 0\\ 2& 2& 2& 1& 0\\ 2& 3& 2& 1& 0\\ 2& 2& 2& 1& 0\\ 1& 1& 1& 1& 0\end{array}\right).$

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

$S^{𝖳}AS=\left(\begin{array}{ccccc}2& 2& 2& 1& 0\\ 2& 3& 2& 1& 0\\ 2& 2& 2& 1& 0\\ 1& 1& 1& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right).$

See also

• Clock and shift matrices
• Nilpotent matrix
• Subshift of finite type

References

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External links

• Shift Matrix - entry in the Matrix Reference Manual