Difference between revisions of "Manuals/calci/SHIFT"
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<math>U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1}</math>. | <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. | where <math>\delta_{ij}</math> is the Kronecker delta symbol. | ||
| − | *For example, the 5×5 shift matrices are | + | *For example, the 5×5 shift matrices are |
<math>U_5=\begin{pmatrix} | <math>U_5=\begin{pmatrix} | ||
0 & 1 & 0 & 0 & 0 \\ | 0 & 1 & 0 & 0 & 0 \\ | ||
| Line 28: | Line 28: | ||
\end{pmatrix}</math> | \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. | *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") | ||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | | 0 || 1 || 0 | ||
| + | |- | ||
| + | | 0 || 0 || 1 | ||
| + | |- | ||
| + | | 0 || 0 || 0 | ||
| + | |} | ||
| + | *2.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== | ||
Revision as of 11:33, 4 May 2015
MATRIX("SHIFT",order)
- is the size of the Shift matrix.
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 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 component of U and L are:
are equal to the matrix A shifted one position down, up left, right, and down along the main diagonal respectively.
component of U and L are:.
.
where is the Kronecker delta symbol.
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.
Examples
- 1.MATRIX("shift")
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 0 | 0 | 0 |
- 2.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 |