Shift matrix

In mathematics, a shift matrix is a binary matrix with ones on the superdiagonal or subdiagonal, and zeroes elsewhere. A shift matrix with ones on the superdiagonal is an upper shift matrix; the alternative subdiagonal case is unsurprisingly known as a lower shift matrix. Clearly, the transpose of a lower shift matrix is an upper shift matrix and vice versa.

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

Formally, if we define n by n matrices L and U as lower and upper shift matrices respectively, the following conditions are satisfied.

• det(L) = det(U) = 0

• trace(L) = trace(U) = 0

• rank(L) = rank(U) = n−1

• Lⁿ = Uⁿ = 0

• L^T = U; U^T = L

• LU and UL are both idempotent, symmetric, and have the same rank as U and L

• L^n-aU^n-a + L^aU^a = U^n-aL^n-a + U^aL^a = I (the identity matrix), for any integer a between 0 and n inclusive

Examples

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

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

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

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