v2010.10.26 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 53Isomorphic vector spaces are characterized by preservation of adjacency;id est, if v and w are points connected by a line segment in one vectorspace, then their images will be connected by a line segment in the other.Two Euclidean bodies may be considered isomorphic if there exists anisomorphism, of their vector spaces, under which the bodies correspond.[367,I.1] Projection (E) is not an isomorphism, Figure 17 for example;hence, perfect reconstruction (inverse projection) is generally impossiblewithout additional information.When Z =Y ∈ R p×k in (38), Frobenius’ norm is resultant from vectorinner-product; (confer (1686))‖Y ‖ 2 F = ‖ vec Y ‖2 2 = 〈Y , Y 〉 = tr(Y T Y )= ∑ i,jY 2ij = ∑ iλ(Y T Y ) i = ∑ iσ(Y ) 2 i(44)where λ(Y T Y ) i is the i th eigenvalue of Y T Y , and σ(Y ) i the i th singularvalue of Y . Were Y a normal matrix (A.5.1), then σ(Y )= |λ(Y )|[393,8.1] thus‖Y ‖ 2 F = ∑ iλ(Y ) 2 i = ‖λ(Y )‖ 2 2 = 〈λ(Y ), λ(Y )〉 = 〈Y , Y 〉 (45)The converse (45) ⇒ normal matrix Y also holds. [202,2.5.4]Frobenius’ norm is the Euclidean norm of vectorized matrices. Becausethe metrics are equivalent, for X ∈ R p×k‖ vec X −vec Y ‖ 2 = ‖X −Y ‖ F (46)and because vectorization (37) is a linear bijective map, then vector spaceR p×k is isometrically isomorphic with vector space R pk in the Euclidean senseand vec is an isometric isomorphism of R p×k . Because of this Euclideanstructure, all the known results from convex analysis in Euclidean space R ncarry over directly to the space of real matrices R p×k .2.2.1.1 Injective linear operatorsInjective mapping (transformation) means one-to-one mapping; synonymouswith uniquely invertible linear mapping on Euclidean space.Linear injective mappings are fully characterized by lack of nontrivialnullspace.