Manuals/calci/SHIFT

MATRIX("SHIFT",order)

• $order$ is the size of the Shift matrix.

Description

• This function returns shift matrix of order 3.
• 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.
• Let Z is a shift matrix , then ZA, ZTA, AZ, AZT, ZAZT are equal to the matrix A shifted one position down, up left, right, and down along the main diagonal respectively.
• 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}, \quad L_{ij} = \delta_{i,j+1},where \delta_{ij} is the Kronecker delta symbol.

• For example, the 5×5 shift matrices are:
• 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.