v2009.01.01 - Convex Optimization

2.2. VECTORIZED-MATRIX INNER PRODUCT 47
2.2.1 Frobenius’
2.2.1.0.1 Definition. Isomorphic.
An isomorphism of a vector space is a transformation equivalent to a linear
bijective mapping. The image and inverse image under the transformation
operator are then called isomorphic vector spaces.
△
Isomorphic vector spaces are characterized by preservation of adjacency;
id est, if v and w are points connected by a line segment in one vector space,
then their images will also be connected by a line segment. Two Euclidean
bodies may be considered isomorphic if there exists an isomorphism, of their
vector spaces, under which the bodies correspond. [320,I.1] Projection (E)
is not an isomorphism, for example; hence, perfect reconstruction (inverse
projection) is generally impossible without additional information.
When Z =Y ∈ R p×k in (31), Frobenius’ norm is resultant from vector
inner-product; (confer (1566))
‖Y ‖ 2 F = ‖ vec Y ‖2 2 = 〈Y , Y 〉 = tr(Y T Y )
= ∑ i,j
Y 2
ij = ∑ i
λ(Y T Y ) i = ∑ i
σ(Y ) 2 i
(37)
where λ(Y T Y ) i is the i th eigenvalue of Y T Y , and σ(Y ) i the i th singular
value of Y . Were Y a normal matrix (A.5.2), then σ(Y )= |λ(Y )|
[344,8.1] thus
‖Y ‖ 2 F = ∑ i
λ(Y ) 2 i = ‖λ(Y )‖ 2 2 = 〈λ(Y ), λ(Y )〉 = 〈Y , Y 〉 (38)
The converse (38) ⇒ normal matrix Y also holds. [176,2.5.4]
Because the metrics are equivalent
‖ vec X −vec Y ‖ 2 = ‖X −Y ‖ F (39)
and because vectorization (30) is a linear bijective map, then vector space
R p×k is isometrically isomorphic with vector space R pk in the Euclidean sense
and vec is an isometric isomorphism of R p×k . 2.12 Because of this Euclidean
structure, all the known results from convex analysis in Euclidean space R n
carry over directly to the space of real matrices R p×k .
2.12 Given matrix A, its range R(A) (2.5) is isometrically isomorphic with its vectorized
range vec R(A) but not with R(vec A).