Difference between revisions of "Manuals/calci/SHIFT"

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<div style="font-size:30px">'''SHIFT'''</div><br/>
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<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 \\
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0 & 0 & 0 & 1 & 0 \\
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0 & 0 & 0 & 0 & 1 \\
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0 & 0 & 0 & 0 & 0
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\end{pmatrix}</math>
 +
<math>L_5 = \begin{pmatrix}
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0 & 0 & 0 & 0 & 0  \\
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1 & 0 & 0 & 0 & 0 \\
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0 & 1 & 0 & 0 & 0 \\
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0 & 0 & 1 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0
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\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==
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*1.MATRIX("shift") = 0
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*2.MATRIX("shift",3)
 +
{| class="wikitable"
 +
|-
 +
| 0 || 1 || 0
 +
|-
 +
| 0 || 0 || 1
 +
|-
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| 0 || 0 || 0
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|}
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*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]]
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*[[Manuals/calci/CONFERENCE| CONFERENCE]]
 +
*[[Manuals/calci/TRIANGULAR| TRIANGULAR]]
 +
 
 +
==References==
 +
*[http://en.wikipedia.org/wiki/Shift_matrix Shift Matrix]

Latest revision as of 01:39, 26 October 2015

MATRIX("SHIFT",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 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:
.

where 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
  • 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

References

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