／var／log marcus chiu

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Algebra - Subfields

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Linear Algebra - Operations／Operators

Outer Product

Created on Sep 13, 2021 · Last Modified on May 23, 2026

Outer Product

the outer product of two vectors 𝑣 and 𝑢 is their tensor product 𝑣⊗𝑢
- if the vectors are given as coordinate columns, then the outer product is the matrix 𝑣𝑢^T
- if vector 𝑣 has dimension 𝑛 and vector 𝑢 has dimension 𝑚, then their outer product is an 𝑛×𝑚 matrix
is the “expanded form” of the geometric product
is related to wedge product

Outer Product - Matrix Definition

Given two vectors

$v = v_{1} v_{2} ⋮ v_{n}, u = u_{1} u_{2} ⋮ u_{m}$

their outer product is defined as:

$v \otimes u = v u^{T} = v_{1} u_{1} v_{2} u_{1} ⋮ v_{n} u_{1} v_{1} u_{2} v_{2} u_{2} ⋮ v_{n} u_{2} \dots \dots ⋱ \dots v_{1} u_{m} v_{2} u_{m} ⋮ v_{n} u_{m}$

Outer Product - Expression Definition

Since a matrix can be represented as a sum of its basis:

$⎩ ⎨ ⎧ 10 ⋮ 0 00 ⋮ 0 \dots \dots ⋱ \dots 00 ⋮ 0, 00 ⋮ 0 10 ⋮ 0 \dots \dots ⋱ \dots 00 ⋮ 0, ..., 00 ⋮ 0 00 ⋮ 0 \dots \dots ⋱ \dots 10 ⋮ 0, ..., ..., 00 ⋮ 0 00 ⋮ 0 \dots \dots ⋱ \dots 00 ⋮ 1, ⎭ ⎬ ⎫$

Each element in its basis can be represented as 𝐸_𝑖𝑗:

${E_{11}, E_{12}, ..., E_{1 m}, ......, E_{n 1}, E_{n 2}, ..., E_{nm}}$

Thus the outer product is expressed as:

$v \otimes u = v u^{T} = v_{1} u_{1} E_{11} + v_{1} u_{2} E_{12} + ... + v_{1} u_{m} E_{1 m} + v_{2} u_{1} E_{21} + v_{2} u_{2} E_{22} + ... + v_{2} u_{m} E_{2 m} + ... v_{n} u_{1} E_{n 1} + v_{n} u_{2} E_{n 2} + ... + v_{n} u_{m} E_{nm}$