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In linear algebraan outer product is the tensor product of two coordinate vectorsa special case of the Kronecker product product name binary matrix programming matrices. The outer product for general tensors is also called the tensor product.
The outer product name binary matrix programming contrasts with the dot productwhich takes as input a pair of coordinate vectors and produces a scalar.
For complex vectors, it is customary to use the conjugate transpose of v denoted v H:. The inner product is the trace of the outer product. If u and v are both nonzero then the outer product matrix uv T always has matrix rank 1, as can be easily seen by multiplying it with a vector x:. Namely, matrix A is obtained product name binary matrix programming multiplying each element of u by the complex conjugate of each element of v.
The outer product on tensors is typically referred to as the tensor product. Given a tensor a of order q with dimensions i 1For example, if Product name binary matrix programming is of order 3 with dimensions 3, 5, 7 and B is of order 2 with dimensions 10,their outer product c is of order 5 with dimensions 3, 5, 7, 10, To understand the matrix definition of outer product in terms of the definition of tensor product:. This scalar in turn is multiplied by x to give as the final result product name binary matrix programming element of the space V.
If V and W are finite-dimensional, then the space of all linear transformations from W to Vdenoted Hom WVis generated by such outer products; in fact, the rank of a matrix is the minimal number of such outer products needed to express it as a sum this is the tensor rank of a matrix. This is syntactically represented in various ways: These often generalize to multi-dimensional arguments, and more than two arguments. The outer product is useful in computing physical quantities e.
It is also useful in statistical analysis for computing the covariance and auto-covariance matrices for two vector-valued random variables. From Wikipedia, the free encyclopedia. This article is about the product of tensors. For "outer product" in geometric algebra, see Geometric algebra. Not to be confused with Exterior product. Encyclopaedia of Physics 2nd ed. Mathematical Methods for Physics and Engineering. Cross product Triple product Seven-dimensional cross product.