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 can be written as an ‘‘outer product’’, or dyad:
where . |
The interpretation of the corresponding linear map for a rank-one matrix is that the output is always in the direction , with coefficient of proportionality a linear function of .
We can always scale the vectors and in order to express as
where , , with and .
The interpretation for the expression above is that the result of the map for a rank-one matrix can be decomposed into three steps:
- we project on the -axis, getting a number ;
- we scale that number by the positive number ;
- we lift the result (which is the scalar to get a vector proportional to .