Difference between revisions of "Manuals/calci/SHIFT"
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(Created page with "<div style="font-size:30px">'''SHIFT'''</div><br/>") |
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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 ZA, ZTA, AZ, AZT, ZAZT 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 | ||
| + | U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1},where \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. | ||
Revision as of 10:03, 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 ZA, ZTA, AZ, AZT, ZAZT 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
U_{ij} = \delta_{i+1,j}, \quad L_{ij} = \delta_{i,j+1},where \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.