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Recall that the rank of a matrix is the dimension of its range. A rank-one matrix is a matrix with rank equal to one. Such matrices are also called dyads.
We can express any rank-one matrix as an outer product.
Theorem: outer product representation of a rank-one matrix
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Every rank-one matrix $A \in \mathbb{R}^{m \times n}$ can be written as an ‘‘outer product’’, or dyad: $$ where $p \in \mathbb{R}^{m}, \; q \in \mathbb{R}^{n}$. |
The interpretation of the corresponding linear map $x \rightarrow y = Ax$ for a rank-one matrix $A$ is that the output $y$ is always in the direction $p$, with coefficient of proportionality a linear function of $x: x \rightarrow q^Tx$.
We can always scale the vectors $p$ and $q$ in order to express $A$ as
$$A = \sigma u v^T,$$
where $u \in \mathbb{R}^{m}$, $v \in \mathbb{R}^{n}$, with $\|u\|_2=\|u\|_2=1,$ and $\sigma > 0$.
The interpretation for the expression above is that the result of the map $x \rightarrow y = Ax$ for a rank-one matrix $A$ can be decomposed into three steps:
- we project $x$ on the $v$-axis, getting a number $v^Tx$;
- we scale that number by the positive number $\sigma$;
- we lift the result (which is the scalar $\sigma(v^Tx))$ to get a vector proportional to $u$.
