v2009.01.01 - Convex Optimization

50 CHAPTER 2. CONVEX GEOMETRY
2.2.1.2 Noninjective linear operators
Mappings in Euclidean space created by noninjective linear operators can
be characterized in terms of an orthogonal projector (E). Consider
noninjective linear operator Tx =Ax : R n → R m representing fat matrix
A∈ R m×n (m< n). What can be said about the nature of this m-dimensional
mapping?
Concurrently, consider injective linear operator Py=A † y : R m → R n
where R(A † )= R(A T ). P(Ax)= PTx achieves projection of vector x on
the row space R(A T ). (E.3.1) This means vector Ax can be succinctly
interpreted as coefficients of orthogonal projection.
Pseudoinverse matrix A † is skinny and full rank, so operator Py is a linear
bijection with respect to its range R(A † ). By Definition 2.2.1.0.1, image
P(T B) of projection PT(B) on R(A T ) in R n must therefore be isomorphic
with the set of projection coefficients T B = {Ax |x∈ B} in R m and have the
same affine dimension by (42). To illustrate, we present a three-dimensional
Euclidean body B in Figure 15 where any point x in the nullspace N(A)
maps to the origin.
2.2.2 Symmetric matrices
2.2.2.0.1 Definition. Symmetric matrix subspace.
Define a subspace of R M×M : the convex set of all symmetric M×M matrices;
S M ∆ = { A∈ R M×M | A=A T} ⊆ R M×M (43)
This subspace comprising symmetric matrices S M is isomorphic with the
vector space R M(M+1)/2 whose dimension is the number of free variables in a
symmetric M ×M matrix. The orthogonal complement [287] [215] of S M is
S M⊥ ∆ = { A∈ R M×M | A=−A T} ⊂ R M×M (44)
the subspace of antisymmetric matrices in R M×M ; id est,
S M ⊕ S M⊥ = R M×M (45)
where unique vector sum ⊕ is defined on page 708.
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