Inner and outer products

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Outer product of a pair of vectors is a matrix. But inner product of a pair of vectors is a scalar.

Inner product

The inner or dot product of two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is defined as:

<math>\mathbf{a}\cdot \mathbf{b} = \sum_{i=1}^n a_ib_i = a_1b_1 + a_2b_2 + \cdots + a_nb_n </math>

where Σ denotes summation notation and n is the dimension of the vector space.

Outer product

The outer or tensor product uv is equivalent to a matrix multiplication uvT, provided that u is represented as a m × 1 column vector and v as a n × 1 column vector (which makes vT a row vector).[১] For instance, if m = 4 and n = 3, then

<math>\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^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}.</math>

References

  1. Linear Algebra (4th Edition), S. Lipcshutz, M. Lipson, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-154352-1