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 :th component of U and L are
where is the Kronecker delta symbol.
For example, the 5×5 shift matrices are
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 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:
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
where each of the blocks S1, S2, ..., Sr is a shift matrix.

Examples

Then
Clearly there are many possible permutations. For example, is equal to the matrix A shifted up and left along the main diagonal.