By Dieudonne J.

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4] Frobenius norm is the Euclidean norm of vectorized matrices. 13 of Rp×k . Because of this Euclidean structure, all the known results from convex analysis in Euclidean space Rn carry over directly to the space of real matrices Rp×k . 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. 1 Definition. Isometric isomorphism.

2]) transformation vectorization. For example, the vectorization of Y = [ y1 y2 · · · yk ] ∈ Rp×k [152] [305] is y1 y2 . ∈ Rpk vec Y (32) .. 2] Y,Z tr(Y TZ) = vec(Y )T vec Z (33) 50 CHAPTER 2. 4]. The adjoint operation AT on a matrix can therefore be defined in like manner: Y , ATZ AY , Z (35) Take any element C1 from a matrix-valued set in Rp×k , for example, and consider any particular dimensionally compatible real vectors v and w . 1 Example. Application of inverse image theorem.

VECTORIZED-MATRIX INNER PRODUCT 55 with the set of projection coefficients T B = {Ax | x ∈ B} in Rm and have the same affine dimension by (44). To illustrate, we present a three-dimensional Euclidean body B in Figure 19 where any point x in the nullspace N (A) maps to the origin. 1 Definition. Symmetric matrix subspace. Define a subspace of RM ×M : the convex set of all symmetric M×M matrices; SM A ∈ RM ×M | A = AT ⊆ RM ×M (45) This subspace comprising symmetric matrices SM is isomorphic with the vector space RM (M +1)/2 whose dimension is the number of free variables in a symmetric M ×M matrix.