Difference between revisions of "Determinant"

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[[category:Matrix Functions]]
 
[[category:Matrix Functions]]
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[[Category:Doc Status C]] <!-- For Lumina use, do not change -->
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[[category:Array-reducing functions]]
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[[Category:Doc Status D]] <!-- For Lumina use, do not change -->
  
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Determinant(C,I,J)
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==Determinant(c, I, J)==
  
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The determinant of a matrix.
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Computes the determinant of a square matrix, «c». Indexes «<code>I</code>» and «<code>J</code>» must be the same length, so that the matrix is square.
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The determinant of a matrix is a useful value that is used heavily in linear algebra and matrix applications. The [[Invert|inverse]] of a matrix exists and is unique if and only if the determinant is non-zero. 
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Geometrically, the determinant can be viewed as the N-dimensional volume of the N-dimensional parallelepiped formed from the vectors in matrix «c». A parallelepiped in the N-dimensional generalization of a parallelogram (a 2-D parallelepiped is a parallelogram). Each vertex of the parallelepiped is obtained by taking a subset of the slices along «<code>J</code>» and adding them together. The empty subset is the origin. There are <code>2^N</code> subsets, where <code>N=[[Size]](J)</code>. 
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== Examples ==
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The determinant of the identity matrix is 1.
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:<code>Determinant(I=J,I,J) &rarr; 1</code>
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With <code>c</code> &rarr; [[image:Determinant_example1.png]],
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:<code>Determinant(c,I,J)</code> &rarr; 15
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With <code>c</code> &rarr; [[image:Determinant_example2.png]],
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:<code>Determinant(c,I,J)</code> &rarr; -114 - 68j
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:<code>Variable Matrix :=</code>
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:{| class="wikitable"
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!
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! colspan="5" |M &#9654;
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|-
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!L &#9660;
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!1
 +
!2
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!3
 +
!4
 +
!5
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|-
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!1
 +
|6
 +
|2
 +
|6
 +
|3
 +
|1
 +
|-
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!2
 +
|2
 +
|4
 +
|3
 +
|1
 +
|3
 +
|-
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!3
 +
|6
 +
|3
 +
|9
 +
|3
 +
|4
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|-
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!4
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|3
 +
|1
 +
|3
 +
|8
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|4
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|-
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!5
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|1
 +
|3
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|4
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|4
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|7
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|}
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:<code>Determinant(Matrix,L,M) &rarr; 359</code>
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==See Also==
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* [[MatrixMultiply]]
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* [[Decompose]]
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* [[Matrix functions]]
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* [[:Category:Matrix Functions]]

Latest revision as of 18:51, 24 March 2016


Determinant(c, I, J)

Computes the determinant of a square matrix, «c». Indexes «I» and «J» must be the same length, so that the matrix is square.

The determinant of a matrix is a useful value that is used heavily in linear algebra and matrix applications. The inverse of a matrix exists and is unique if and only if the determinant is non-zero.

Geometrically, the determinant can be viewed as the N-dimensional volume of the N-dimensional parallelepiped formed from the vectors in matrix «c». A parallelepiped in the N-dimensional generalization of a parallelogram (a 2-D parallelepiped is a parallelogram). Each vertex of the parallelepiped is obtained by taking a subset of the slices along «J» and adding them together. The empty subset is the origin. There are 2^N subsets, where N=Size(J).

Examples

The determinant of the identity matrix is 1.

Determinant(I=J,I,J) → 1

With c → Determinant example1.png,

Determinant(c,I,J) → 15

With c → Determinant example2.png,

Determinant(c,I,J) → -114 - 68j


Variable Matrix :=
M ▶
L ▼ 1 2 3 4 5
1 6 2 6 3 1
2 2 4 3 1 3
3 6 3 9 3 4
4 3 1 3 8 4
5 1 3 4 4 7
Determinant(Matrix,L,M) → 359

See Also

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