:for "outer product" in geometric algebra, see exterior algebra. in linear algebra, the term outer product typically refers to the tensor product of two vectors. the result of applying the outer product to a pair of coordinate vectors is a matrix. the name contrasts with the inner product, which takes as input a pair of vectors and produces a scalar. the outer product of vectors can be also regarded as a special case of the kronecker product of matrices. some authors use the expression "outer product of tensors" as a synonym of "tensor product". the outer product is also a higher-order function in some computer programming languages such as r, apl, and mathematica. ==definition (matrix multiplication)==  the outer product  is equivalent to a matrix multiplication uvt, provided that u is represented as a  column vector and v as a  column vector (which makes vt a row vector).linear algebra (4th edition), s. lipcshutz, m. lipson, schaumâs outlines, mcgraw hill (usa), 2009, isbn 978-0-07-154352-1 for instance, if  and , then :\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^\mathrm{t} = \begin{bmatrix}u_1 \\ u_2 \\ u_3 \\ u_4\end{bmatrix} \begin{bmatrix}v_1 & v_2 & v_3\end{bmatrix} = \begin{bmatrix}u_1v_1 & u_1v_2 & u_1v_3 \\ u_2v_1 & u_2v_2 & u_2v_3 \\ u_3v_1 & u_3v_2 & u_3v_3 \\ u_4v_1 & u_4v_2 & u_4v_3\end{bmatrix}. or in index notation: :(\mathbf{u} \mathbf{v}^\mathrm{t})_{ij}=u_iv_j for complex vectors, it is customary to use the conjugate transpose of v (denoted vh): :\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^\mathrm{h}. ===contrast with inner product=== if , then one can take the matrix product the other way, yielding a scalar (or  matrix): :\left\langle \mathbf{u}, \mathbf{v}\right\rangle = \mathbf{u}^\mathrm{t} \mathbf{v} which is the standard inner product for euclidean vector spaces, better known as the dot product. the inner product is the trace of the outer product. ===rank of an outer product=== if u and v are both nonzero then the outer product matrix uvt always has matrix rank 1, as can be easily seen by multiplying it with a vector x: :(\mathbf{u} \mathbf{v}^\mathrm{t}) \mathbf{x} = \mathbf{u} (\mathbf{v}^\mathrm{t} \mathbf{x}) which is just a scalar vtx multiplied by a vector u. ("matrix rank" should not be confused with "tensor order", or "tensor degree", which is sometimes referred
Bilinear_operators Binary_operations Higher-order_functions