v2009.01.01 - Convex Optimization

48 CHAPTER 2. CONVEX GEOMETRY
T
R 2 R 3
dim domT = dim R(T)
Figure 14: Linear injective mapping Tx=Ax : R 2 →R 3 of Euclidean
body remains two-dimensional under mapping representing skinny full-rank
matrix A∈ R 3×2 ; two bodies are isomorphic by Definition 2.2.1.0.1.
2.2.1.1 Injective linear operators
Injective mapping (transformation) means one-to-one mapping; synonymous
with uniquely invertible linear mapping on Euclidean space. Linear injective
mappings are fully characterized by lack of a nontrivial nullspace.
2.2.1.1.1 Definition. Isometric isomorphism.
An isometric isomorphism of a vector space having a metric defined on it is a
linear bijective mapping T that preserves distance; id est, for all x,y ∈dom T
‖Tx − Ty‖ = ‖x − y‖ (40)
Then the isometric isomorphism T is a bijective isometry.
Unitary linear operator Q : R k → R k representing orthogonal matrix
Q∈ R k×k (B.5) is an isometric isomorphism; e.g., discrete Fourier transform
via (747). Suppose T(X)= UXQ , for example. Then we say the Frobenius
norm is orthogonally invariant; meaning, for X,Y ∈ R p×k and dimensionally
compatible orthonormal matrix 2.13 U and orthogonal matrix Q
‖U(X −Y )Q‖ F = ‖X −Y ‖ F (41)
2.13 Any matrix U whose columns are orthonormal with respect to each other (U T U = I);
these include the orthogonal matrices.
△