Manuals/calci/SHIFT

MATRIX("SHIFT",order)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ZA, Z^TA, AZ, AZ^T, ZAZ^T} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (i,j)^th} component of U and L are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{ij} = \delta_{i+1,j},  L_{ij} = \delta_{i,j+1}}
.

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.