Mixed product

In geometry, the produces mixed is the name which the determinant within an Euclidean framework takes. Its absolute value is interpreted like the volume of a Parallélotope.

For the mixed product in an Euclidean space of dimension three, to see the article vectorial geometry.

Definition

That is to say E a Euclidean Space directed of dimension N. That is to say B a direct orthonormal base of E . The mixed product of N vectors of E is defined by (x_1,…, x_n) \ mapsto = \ det_B (x_1,…, x_n)

It does not depend on the direct orthonormal base B selected.

Demonstration : the endomorphisms which send direct orthonormal base on direct orthonormal basis are the orthogonal automorphisms of determinant 1. The determinant of a family of vectors x1,… xn in two direct orthonormal bases thus has the same value.

The mixed product is null if and only if the family of the xi is dependant, strictly positive if and only it constitutes a direct base, 1 is worth if and only if it constitutes it also a direct orthonormal base.

It checks the inequality of Hadamard

x_n \ Leq \ prod \ limits_ {i=1} ^n \|x_i \|
When the vectors train a free family, there is equality if and only if this family is orthogonal. In other words, the lengths on the sides being given, the Parallélotope right is that which has largest volume.

For the manufacture of particular vectors (with coefficients 1 and -1) checking the case equality to see Matrix of Hadamard.

Volumes of parallélotopes of size lower than N

In an Euclidean space, and even in a real Space préhilbertien of unspecified size, the determinants also allow the calculation of volumes of the parallélotopes of any dimension finished in the form of matrices and determinants of Gram.

They are this time not directed volumes, and it is not possible to give a directed version of it.

Bond of the mixed product with the product external and the duality of Hodge

By Duality of Hodge, it is possible to pass from the 0-vector 1 to a N - vector of the form produces external of the vectors of a direct orthonormal base e1,…, en . The product external of unspecified N vectors is thus written

x_1 \ wedge x_2 \ wedge \ dowries \ wedge x_n = x_n.*1

It is also possible to see the application produces mixed like a form N - linear dual of the 0-form 1

= {\ rm D} e_1 \ wedge {\ rm D} e_2 \ wedge \ dowries \ wedge {\ rm D} e_n=*1

Application: general standard of the vector product

With the same notations the vector product \ bigwedge (x_1, \ dowries, x_n) of n-1 vectors of E , x1,…, xn-1 , is defined by

\ forall X \ in E, \ qquad = \ left (\ bigwedge (x_1, \ dowries, x_n)|X \ right)

The application product vector is (n-1) - linear alternate. The vector product is cancelled if and only if the family is dependant.

The coordinates of the vector product are given by

\ bigwedge (x_1, \ cdots, x_ {n-1}) =
\begin{vmatrix} x_1 {} ^1 & \ cdots &x_1 {} ^ {N} \ \ \ vdots & \ ddots & \ vdots \ \ x_ {n-1} {} ^1 & \ cdots &x_ {n-1} {} ^ {N} \ \ \ mathbf {E} _1 & \ cdots & \ mathbf {E} _ {N} \end{vmatrix} by noting ei vectors of the direct orthonormal base. In other words the coordinates of the vector product are cofacteurs of this matrix.

By Duality of Hodge, the vector product and the produce external of n-1 vectors correspond

\ bigwedge (x_1, \ cdots, x_ {n-1}) =* \ left (x_1 \ wedge x_2 \ wedge \ dowries \ wedge x_ {n-1} \ right)
This constitutes an alternative definition of the vector product.

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