Search results
Results from the WOW.Com Content Network
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.
the outer product of two column vectors and is denoted and defined as or , where means transpose, the tensor product of two vectors a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } is denoted a ⊗ b {\displaystyle \mathbf {a} \otimes \mathbf {b} } ,
The tensor product, outer product and Kronecker product all convey the same general idea. The differences between these are that the Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its intrinsic definition .
The product in the algebra is called the geometric product, and the product in the contained exterior algebra is called the exterior product (frequently called the wedge product or the outer product [d]).
Consider now the exterior product of and : = (+) (+) = + + + = (), where the first step uses the distributive law for the exterior product, and the last uses the fact that the exterior product is an alternating map, and in particular = (). (The fact that the exterior product is an alternating map also forces = =
The definition of matrix product requires that the entries belong to a semiring, and does not require multiplication of elements of the semiring to be commutative. In many applications, the matrix elements belong to a field, although the tropical semiring is also a common choice for graph shortest path problems. [15]
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
The outer product is defined for different dimensions, while the inner product requires the same dimension. If the dimensions are the same, then the inner product is the trace of the outer product (trace only being properly defined for square matrices). In an informal summary: "inner is horizontal times vertical and shrinks down, outer is ...