Definition:Outer Product

Definition

Let $\mathbf u = \tuple {u_1, u_2, \ldots, u_m}$ and $\mathbf v = \tuple {v_1, v_2, \ldots, v_n}$ be vectors.

The outer product of $\mathbf u$ and $\mathbf v$ is defined and denoted as:

$\mathbf u \otimes \mathbf v := \begin {bmatrix} u_1 v_1 & u_1 v_2 & \dots & u_1 v_n \\ u_2 v_1 & u_2 v_2 & \dots & u_2 v_n \\ \vdots & \vdots & \ddots & \vdots \\ u_m v_1 & u_m v_2 & \dots & u_m v_n \end {bmatrix}$

That is:

$\sqbrk {u \otimes v}_{i j} := u_i v_j$

If it is understood that $\mathbf u$ and $\mathbf v$ are expressed as column vectors:

$\mathbf u \otimes \mathbf v := \mathbf u \mathbf v^\intercal$

where $\mathbf v^\intercal$ denotes the transpose of $\mathbf v$.

Also known as

The outer product between two vectors is also known as their dyad product.

Also see

• Product of Outer Product with Multiplicand
• Transpose of Outer Product

• Definition:Inner Product

• Results about outer products can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): outer product
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): outer product