Answer to Question #24671 in Linear Algebra for Mikhail Vega

Question

1.a) Prove that the product A = v(w^T) of a nonzero m x 1 column vector v by a nonzero 1 x n
row vector w^T is an m x n matrix of rank 1. [Hint: do a few small examples]
b) Now show that if A is an m x n matrix of rank 1, then there exist a nonzero m x 1 column
vector v and a nonzero 1 x n row vector w^T such that A = v(w^T).

Answer 1

1
Fact& rank(AB)<=min{rank(A), rank(B)}
rank(v * w^t)<=1 =>& rank A = 1

2
If rank A =1 then each column is a multiply of some& column vector v, so A can be expressed as ( av,& bv,& ...& , cv), where a,b,....,c are constants.
Then A=v * (a,b,...,c)^t