Rank-one matrices

L. El Ghaoui

Rank-one matrices

Recall that the rank of a matrix is the dimension of its range. A rank-one matrix is a matrix with a 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

Every rank-one matrix $A \in \mathbb{R}^{m \times n}$ can be written as an ‘‘outer product’’, or dyad

$A=pq^{T},$

where $p \in \mathbb{R}^{m}, q \in \mathbb{R}^{n}$ .

Proof of the theorem.

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 a coefficient of proportionality a linear function of $x$ : $x \rightarrow q^{T}x$ .

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 $||u||_{2}=||v||_{2}=1$ , and $\sigma > 0$ .

The interpretation for the expression above is that the result of the map $x \rightarrow Ax$ for a rank-one matrix $A$ can be decomposed into three steps:

we project $x$ on the $v$ -axis, getting a number $v^{T}x$ ;
we scale that number by the positive number $\sigma$ ;
we lift the result (which is the scalar $\sigma(v^{T}x)$ ) to get a vector proportional to $u$ .

Rank-one matrices

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