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<div style="font-size:30px">'''SHIFT'''</div><br/>
<div style="font-size:30px">'''MATRIX("SHIFT",order)'''</div><br/>
*<math>order</math> 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 <math>ZA, Z^TA, AZ, AZ^T, ZAZ^T</math> 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 <math>(i,j)^{th}</math> component of U and L are:
<math>U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1}</math>.
where <math>\delta_{ij}</math> 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>
<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}</math>
*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)
{| class="wikitable"
|-
| 0 || 1 || 0
|-
| 0 || 0 || 1
|-
| 0 || 0 || 0
|}
*3.MATRIX("shift",7)
{| class="wikitable"
|-
| 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==
*[[Manuals/calci/SIGNATURE| SIGNATURE]]
*[[Manuals/calci/CONFERENCE| CONFERENCE]]
*[[Manuals/calci/TRIANGULAR| TRIANGULAR]]
==References==
*[http://en.wikipedia.org/wiki/Shift_matrix Shift Matrix]