Outer Product

Let $\boldsymbol{A}$ and $\boldsymbol{B}$ be $n \times 1$-matrices.

The matrix product of

$$\boldsymbol{M}= \boldsymbol{A} \boldsymbol{B}^T$$

is called the outer product ([📖LLM21, p. 133]) of $\boldsymbol{A}$ and $\boldsymbol{B}$.

Each entry $\boldsymbol{M}_{ij}$ corresponds to the product of the $i$th entry of $\boldsymbol{A}$ and the $j$th entry of $\boldsymbol{B}$:

$$\boldsymbol{M}_{ij} = A_i B_j$$

Example

$$\begin{pmatrix} a_1 \\ a_2 \end{pmatrix} \begin{pmatrix}b_1 & b_2\end{pmatrix} = \begin{pmatrix} a_1 b_1 & a_1 b_2 \\ a_2 b_1 & a_2 b_2\end{pmatrix}$$

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References

1. [LLM21]: Lay, David and Lay, Steven and McDonald, Judi: Linear Algebra and Its Applications Global Edition (2021), Pearson Deutschland [BibTeX]