v2009.01.01 - Convex Optimization

578 APPENDIX B. SIMPLE MATRICES
B.1 Rank-one matrix (dyad)
Any matrix formed from the unsigned outer product of two vectors,
Ψ = uv T ∈ R M×N (1492)
where u ∈ R M and v ∈ R N , is rank-one and called a dyad. Conversely, any
rank-one matrix must have the form Ψ . [176, prob.1.4.1] Product −uv T is
a negative dyad. For matrix products AB T , in general, we have
R(AB T ) ⊆ R(A) , N(AB T ) ⊇ N(B T ) (1493)
with equality when B =A [287,3.3,3.6] B.1 or respectively when B is
invertible and N(A)=0. Yet for all nonzero dyads we have
R(uv T ) = R(u) , N(uv T ) = N(v T ) ≡ v ⊥ (1494)
where dim v ⊥ =N −1.
It is obvious a dyad can be 0 only when u or v is 0;
Ψ = uv T = 0 ⇔ u = 0 or v = 0 (1495)
The matrix 2-norm for Ψ is equivalent to the Frobenius norm;
‖Ψ‖ 2 = ‖uv T ‖ F = ‖uv T ‖ 2 = ‖u‖ ‖v‖ (1496)
When u and v are normalized, the pseudoinverse is the transposed dyad.
Otherwise,
Ψ † = (uv T ) † vu T
=
(1497)
‖u‖ 2 ‖v‖ 2
B.1 Proof. R(AA T ) ⊆ R(A) is obvious.
R(AA T ) = {AA T y | y ∈ R m }
⊇ {AA T y | A T y ∈ R(A T )} = R(A) by (132)