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In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions n and m , then their outer product is an n × m matrix.
There are two lists of mathematical identities related to vectors: Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
The cross product (also called the vector product or outer product) is only meaningful in three or seven dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as
This product assumes the partitions of the matrices are their columns. In this case m 1 = m, p 1 = p, n = q and for each j: n j = q j = 1. The resulting product is a mp × n matrix of which each column is the Kronecker product of the corresponding columns of A and B. Using the matrices from the previous examples with the columns partitioned:
The dot product takes in two vectors and returns a scalar, while the cross product [a] returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A ...
This reassures the existence of a inner product between the two sets of coordinates i.e. it leads to meaningful interpretations of their spatial relations in a biplot. In practical terms one can think of the standard coordinates as the vertices of the vector space in which the set of principal coordinates (i.e. the respective points) "exists". [17]
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
The exterior algebra is named after Hermann Grassmann, [3] and the names of the product come from the "wedge" symbol and the fact that the product of two elements of is "outside" . The wedge product of k {\displaystyle k} vectors v 1 ∧ v 2 ∧ ⋯ ∧ v k {\displaystyle v_{1}\wedge v_{2}\wedge \dots \wedge v_{k}} is called a blade of degree k ...