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.
[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.
The infinite-dimensional shift matrices are particularly important for the study of ergodic systems.
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 (possibly of different sizes).
is equal to the matrix A shifted up and left along the main diagonal.