9.1 Outer Product of Vectors

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 9.2 Example of Matrix Units, 9.3 Example of Outer Product
Constructions: N/A
Generalizations: N/A

Properties: N/A
Sufficiencies: 4.4 Transpose of a Vector
Questions: N/A

Outer Product of Vectors

Let 3.1 Column Vector $\vec{x} \in R^{m \times 1}$ and 3.4 Row Vector $\vec{y} \in R^{n \times 1}$ . Then, the outer product between $\vec{x}$ and $\vec{y}$ is the $m \times n$ matrix given by

\vec{x} {\vec{y}}^{T} = [\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{m} \end{matrix}] [\begin{matrix} y_{1} & y_{2} & \dots & x_{n} \end{matrix}] = [\begin{matrix} x_{1} y_{1} & x_{1} y_{2} & \dots & x_{1} y_{n} \\ x_{2} y_{1} & x_{2} y_{2} & \dots & x_{2} y_{n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{m} y_{1} & x_{m} y_{2} & \dots & x_{m} y_{n} \end{matrix}]

multiply column vector $\vec{x}$ with the transpose (horizontal orientation) of $\vec{y}$

Notice that row $i$ of the outer product $\vec{x} {\vec{y}}^{T}$ is given by $x_{i} {\vec{y}}^{T}$ for $i = 1, 2, \dots, m$ while column $k$ of the outer product is given by $y_{k} \vec{x}$ for $k = 1, 2, \dots, n$ .

Remark. If $\vec{x}, \vec{y} \neq 0$ , then the resulting matrix is called a rank 1 matrix (rank counts the number of linearly independent columns).

[\begin{matrix} x_{1} y_{1} & x_{1} y_{2} & \dots & x_{1} y_{n} \\ x_{2} y_{1} & x_{2} y_{2} & \dots & x_{2} y_{n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ x_{m} y_{1} & x_{m} y_{2} & \dots & x_{m} y_{n} \end{matrix}]

Let's look at $y$ values of the first column. They are all $y_{1}$ . From this information, we can determine that the first column is a scalar multiple of the other columns, meaning that the columns are 6.4 Linearly Dependent Vectors. Every single column is linearly dependent on the first column, or vice versa.

& There is exactly one linearly dependent column in this matrix, so we call it a rank 1 matrix!
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Remark. The outer product between vectors can be very useful for computing many multiplications between real numbers simultaneously. These matrices are also very helpful for creating new matrices.
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