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Difference between revisions of "Manuals/calci/TRIANGULAR"

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<div style="font-size:30px">'''TRIANGULAR'''</div><br/>
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<div style="font-size:30px">'''MATRIX("TRIANGULAR",order)'''</div><br/>
 +
*<math>order</math> is the size of the Triangular matrix.
 +
 
 +
==Description==
 +
*This function gives a triangular matrix of order 3.
 +
*A square matrix is called triangular matrix if that matrix is an upper triangular matrix or lower triangular matrix.
 +
*A square matrix is called lower triangular if all the entries above the main diagonal are zero.
 +
*Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.
 +
*So a triangular matrix is a special kind of square matrix.
 +
*A triangular matrix is one that is either lower triangular or upper triangular.
 +
*Some matrices, such as the identity matrix, are both upper and lower triangular.
 +
*A matrix is upper and lower triangular simultaneously if and only if it is a diagonal matrix.
 +
*Also lower triangular matrix is called left triangular matrix  and upper triangular matrix is called right triangular matrix.
 +
*Triangular matrices have the following properties:
 +
#The inverse of a triangular matrix is a triangular matrix.
 +
#The product of two triangular matrices is a triangular matrix.
 +
#The determinant of a triangular matrix is the product of the diagonal elements.
 +
#The eigenvalues of a triangular matrix are the diagonal elements.
 +
*In calci, MATRIX("triangular") gives the triangular matrix of order 3.
 +
*MATRIX("uppertriangular") or MATRIX("upper-triangular") gives the upper triangular matrix of oreder 3.
 +
*Also MATRIX("lowertriangular") or MATRIX("lower-triangular") is showing the lower triangular matrix of order 3.
 +
*So in Calci, users can get the different types of triangular matrices with the different orders.
 +
 
 +
 
 +
==Examples==
 +
*1.MATRIX("triangular",3)
 +
{| class="wikitable"
 +
|-
 +
| 15 || -96 || -53
 +
|-
 +
| 0 || 94 || 0
 +
|-
 +
| 0 || 0 || 77
 +
|}
 +
*2.MATRIX("triangular",6)
 +
{| class="wikitable"
 +
|-
 +
| 49 || 0 || 0 || 0 || 0 || 0
 +
|-
 +
| 55 || 93 || 0 || 0 || 0 || 0
 +
|-
 +
| -30 || -42 || 48 || 0 || 0 || 0
 +
|-
 +
| -82 || 48 || -9 || 62 || 0 || 0
 +
|-
 +
| -6 || -37 || -68 || -6 || -7 || 0
 +
|-
 +
| 36 || 62 || 28 || -96 || 18 || 55
 +
|}
 +
*3.MATRIX("uppertriangular")
 +
{| class="wikitable"
 +
|-
 +
| -54 || 24 || 28
 +
|-
 +
| 0 || -9 || 79
 +
|-
 +
| 0 || 0 || 84 
 +
|}
 +
*4.MATRIX("lower-triangular",4)
 +
{| class="wikitable"
 +
|-
 +
| 16 || 0 || 0 || 0
 +
|-
 +
| -66 || -17 || 0 || 0
 +
|-
 +
| 69 || -93 || -6 || 0
 +
|-
 +
|25 || -18 || 12 || 40 
 +
|}
 +
 
 +
==Related Videos==
 +
 
 +
{{#ev:youtube|7-jSS6f_urg|280|center|Triangular Matix}}
 +
 
 +
==See Also==
 +
*[[Manuals/calci/PERSYMMETRIC| PERSYMMETRIC]]
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*[[Manuals/calci/PASCAL| PASCAL]]
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*[[Manuals/calci/HANKEL| HANKEL]]
 +
 
 +
==References==
 +
*[http://en.wikipedia.org/wiki/Triangular_matrix Triangular Matrix]
 +
*[http://mathworld.wolfram.com/UpperTriangularMatrix.html Upper Triangular Matrix]
 +
*[http://mathworld.wolfram.com/LowerTriangularMatrix.html Lower Triangular Matrix]

Latest revision as of 01:44, 26 October 2015

MATRIX("TRIANGULAR",order)


  • is the size of the Triangular matrix.

Description

  • This function gives a triangular matrix of order 3.
  • A square matrix is called triangular matrix if that matrix is an upper triangular matrix or lower triangular matrix.
  • A square matrix is called lower triangular if all the entries above the main diagonal are zero.
  • Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.
  • So a triangular matrix is a special kind of square matrix.
  • A triangular matrix is one that is either lower triangular or upper triangular.
  • Some matrices, such as the identity matrix, are both upper and lower triangular.
  • A matrix is upper and lower triangular simultaneously if and only if it is a diagonal matrix.
  • Also lower triangular matrix is called left triangular matrix and upper triangular matrix is called right triangular matrix.
  • Triangular matrices have the following properties:
  1. The inverse of a triangular matrix is a triangular matrix.
  2. The product of two triangular matrices is a triangular matrix.
  3. The determinant of a triangular matrix is the product of the diagonal elements.
  4. The eigenvalues of a triangular matrix are the diagonal elements.
  • In calci, MATRIX("triangular") gives the triangular matrix of order 3.
  • MATRIX("uppertriangular") or MATRIX("upper-triangular") gives the upper triangular matrix of oreder 3.
  • Also MATRIX("lowertriangular") or MATRIX("lower-triangular") is showing the lower triangular matrix of order 3.
  • So in Calci, users can get the different types of triangular matrices with the different orders.


Examples

  • 1.MATRIX("triangular",3)
15 -96 -53
0 94 0
0 0 77
  • 2.MATRIX("triangular",6)
49 0 0 0 0 0
55 93 0 0 0 0
-30 -42 48 0 0 0
-82 48 -9 62 0 0
-6 -37 -68 -6 -7 0
36 62 28 -96 18 55
  • 3.MATRIX("uppertriangular")
-54 24 28
0 -9 79
0 0 84
  • 4.MATRIX("lower-triangular",4)
16 0 0 0
-66 -17 0 0
69 -93 -6 0
25 -18 12 40

Related Videos

Triangular Matix

See Also

References

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