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 $(i,j)^{th}$ 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},  \quad 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:

<math>U_5=\begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix} <math>L_5 = \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 & 0 & 0 \end{pmatrix}

• 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.