Rank-one matrices

Alicia Y. Tsai; Authors: Laurent EI Ghaoui; Contributors: Phan Hoang, Tony Tin, Ha To Tam, Nguyen Van Kep, Le Dinh Nam, Juliette Decugis, Nguyen Quoc Cuong.; Giuseppe C. Calafiore

Rank-one matrices

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

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.

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 $u$ .

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

Theorem: outer product representation of a rank-one matrix

License

Share This Book