v2009.01.01 - Convex Optimization

B.1. RANK-ONE MATRIX (DYAD) 579
R(v)
0 0
R(Ψ) = R(u)
N(Ψ)= N(v T )
N(u T )
R N = R(v) ⊕ N(uv T )
N(u T ) ⊕ R(uv T ) = R M
Figure 132:
The four fundamental subspaces [289,3.6] of any dyad
Ψ = uv T ∈R M×N . Ψ(x) ∆ = uv T x is a linear mapping from R N to R M . The
map from R(v) to R(u) is bijective. [287,3.1]
When dyad uv T ∈R N×N is square, uv T has at least N −1 0-eigenvalues
and corresponding eigenvectors spanning v ⊥ . The remaining eigenvector u
spans the range of uv T with corresponding eigenvalue
λ = v T u = tr(uv T ) ∈ R (1498)
Determinant is a product of the eigenvalues; so, it is always true that
det Ψ = det(uv T ) = 0 (1499)
When λ = 1, the square dyad is a nonorthogonal projector projecting on its
range (Ψ 2 =Ψ ,E.6); a projector dyad. It is quite possible that u∈v ⊥ making
the remaining eigenvalue instead 0 ; B.2 λ = 0 together with the first N −1
0-eigenvalues; id est, it is possible uv T were nonzero while all its eigenvalues
are 0. The matrix
[ ] 1 [ 1 1]
=
−1
[ 1 1
−1 −1
]
(1500)
for example, has two 0-eigenvalues. In other words, eigenvector u may
simultaneously be a member of the nullspace and range of the dyad.
The explanation is, simply, because u and v share the same dimension,
dimu = M = dimv = N :
B.2 A dyad is not always diagonalizable (A.5) because its eigenvectors are not necessarily
independent.