Rank-one matrices: A representation theorem
We prove the theorem mentioned here:
Theorem: outer product representation of a rank-one matrix
Every rank-one matrix can be written as an ‘‘outer product’’, or dyad:
where . |
Proof: For any non-zero vectors , the matrix is indeed of rank one: if , then
When spans , the scalar spans the entire real line (since ), and the vector spans the subspace of vectors proportional to . Hence, the range of is the line:
which is of dimension 1.
Conversely, if is of rank one, then its range is of dimension one, hence it must be a line passing through . Hence for any there exist a function such that
Using , where is the th vector of the standard basis, we obtain that there exist numbers such that for every :
We can write the above in a single matrix equation:
Now letting and realizing that the matrix is simply the identity matrix, we obtain , as desired.