By William Martin Baker
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Additional resources for Algebraic Geometry: A New Treatise On Analytical Conic Sections
Two Euclidean bodies may be considered isomorphic of there exists an isomorphism of their corresponding ambient spaces. 1] When Z = Y ∈ Rp×k in (30), Frobenius’ norm is resultant from vector inner-product; Y 2 F = 2 2 vec Y = Y,Y Yij2 = = i, j = tr(Y T Y ) λ(Y T Y )i = i σ(Y )2i (35) i 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 . 1] thus Y 2 F λ(Y )2i = λ(Y ) = 2 2 (36) i The converse (36) ⇒ normal matrix Y Because the metrics are equivalent vec X − vec Y 2 also holds.
1 relative interior We distinguish interior from relative interior throughout. 24] and it is always possible to pass to a smaller ambient Euclidean space where a nonempty set acquires an interior. 3]. Given the intersection of convex set C with an affine set A rel int(C ∩ A) = rel int(C) ∩ A (12) If C has nonempty interior, then rel int C = int C . 4 Superfluous mingling of terms as in relatively nonempty set would be an unfortunate consequence. From the opposite perspective, some authors use the term full or full-dimensional to describe a set having nonempty interior.
The ambient space of symmetric matrices SM , the antihollow subspace is nontrivial; ∆ ⊥ SM = h δ 2(A) | A ∈ SM = δ(u) | u ∈ RM ⊆ SM (61) In anticipation of their utility with Euclidean distance matrices (EDMs) in 4, for symmetric hollow matrices we introduce the linear bijective vectorization dvec that is the natural analogue to symmetric matrix vectorization svec (46): for Y = [Yij ] ∈ SM h Y12 Y13 Y23 ∆ √ Y 14 ∈ RM (M −1)/2 dvec Y = 2 (62) Y 24 Y 34 .
Algebraic Geometry: A New Treatise On Analytical Conic Sections by William Martin Baker