Let $\bf{u,v}$ be two column vector in $\mathbb R^n$, which can be represented by $n\times1$ matrix. $\bf u^T v$ is the inner product of $\bf u,v$, then is there meaning for $\bf uv^T$, which is a $n\times n$ matrix? Thanks.
Asked
Active
Viewed 590 times
2 Answers
3
This is called the outer product.
Here's a reason why it may be of interest to you. If ${\bf u}$, ${\bf v} \in \mathbb{R}^n$ are non-zero then ${\bf u}{\bf v}^T$ has rank one, and every rank one matrix can be written in this form (exercise). So not every $n \times n$ matrix can be written as ${\bf u}{\bf v}^T$ for some ${\bf u}, {\bf v} \in \mathbb{R}^n$ unless $n = 1$; in particular, any matrix with rank greater than one. However, we have the following relationship between the rank and the outer product.
An $n \times n$ matrix $A$ has rank $k$ if and only if $A$ can be written as the sum of $k$ outer products but not $k - 1$ outer products.
More generally, we don't need ${\bf u}$ and ${\bf v}$ to have the same number of components. We could have ${\bf u} \in \mathbb{R}^m$ and ${\bf v} \in \mathbb{R}^n$ in which case ${\bf u}{\bf v}^T$ is an $m \times n$ matrix, and this is still called the outer product. The previous result carries over to this case as well.
Michael Albanese
- 99,526