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 $ZA,Z^{T}A,AZ,AZ^{T},ZAZ^{T}$ 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

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_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}} Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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.

Examples

• 1.MATRIX("shift") = 0
• 2.MATRIX("shift",3)

0	1	0
0	0	1
0	0	0

• 3.MATRIX("shift",7)

0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1
0	0	0	0	0	0	0

See Also

• SIGNATURE
• CONFERENCE
• TRIANGULAR

References

• Shift Matrix