Appendix F Notation, Definitions, Glossary b scalar or column vector (italic abcdefghijklmnopqrstuvwxyz) th th bi i entry of vector b =[bi , i=1 ...n] or i b vector from a list bj , j =1 ...n or ith iterate of vector b { } th th bi:j or b(i:j) : truncated vector comprising i through j entry of vector b (294) th th bk(i:j) or bi:j,k : truncated vector comprising i through j entry of vector bk bT vector transpose or row vector H T b complex conjugate transpose b∗ A matrix (italic ABCDEFGHIJKLMNOPQRSTUVWXY Z) T A Matrix transpose [Aij ] [Aji] is a linear operator. Regarding A as a linear← operator, AT is its adjoint. T 1 T T 1 A− matrix transpose of inverse; and vice versa, A− = A − (confer p.489 no.47) AT1 first of various transpositions of a cubix or quartix¡ ¢ A (p.¡ 555¢ , p.559) 1 A− inverse of matrix A A† Moore-Penrose pseudoinverse of matrix A (§E) 1 2 3 1 th ··· 2 T Aij or A(i,j) : ij entry of matrix A = or rank-1 matrix aiaj (§4.10) ··· 3 ··· A(i,j) A is a function of i and j th th th Ai i matrix from a set or i principal submatrix (1329) or i iterate of A th A(i , :) or Ai: : i row of matrix A th A(: ,j) or A:j : j column of matrix A [185, §1.1.8] th th th th A(i:j , k :ℓ) or Ai:j, k:ℓ : submatrix; i through j row and k through ℓ column of A √ positive square root √◦ x entrywise positive square root of vector x Dattorro, Convex Optimization Euclidean Distance Geometry 2ε, εβoo, v2018.09.21. 629 M 630 APPENDIX F. NOTATION, DEFINITIONS, GLOSSARY √ℓ positive ℓth root A1/2 and √A A1/2 is any matrix such that A1/2A1/2 =A . n n § For A S , √A S is unique and √A√A =A . [59, §1.2] ( A.5.1.4) ∈ + ∈ + ◦ √D = [ dij ] absolute distance matrix (1517) or Hadamard positive square root: D = √◦ D √◦ D p ◦ thin a skinny matrix; meaning, more rows than columns: When there are more equations than unknowns, we say that the system Ax = b is overdetermined. [185, §5.3] wide a fat matrix; meaning, more columns than rows: underdetermined £ ¤ hollow matrix having 0 main diagonal some set (calligraphic ) A ABCDEFGHIJKLMNOPQRST UVWXYZ A set of vectors or matrices (blackboard ABCDEFGHIJKLMNOPQRSTUVWXYZ) F discrete Fourier transform (932) (Euler Fraktur ABCDEFGHIJKLMNOPQRSTUVWXYZ) ( A) smallest face (176) that contains element A of set F C ∋ C ( ) generators (§2.13.4.2.1) of set ; G K any collection of points and directionsK whose hull constructs K ν level set (571) Lν sublevel set (575) Lν ν superlevel set (669) L L Lagrangian (516) E member of elliptope (1258) parametrized by scalar t Et elliptope (1237) E E elementary matrix Eij member of standard orthonormal basis for symmetric (62) or symmetric hollow (78) matrices id est from the Latin meaning that is e.g exempli gratia, from the Latin meaning for sake of example sic from the Latin meaning so or thus or in this manner; something meant as written videlicet from the Latin meaning it is permitted to see ibidem from the Latin meaning in the same place no. number, from the Latin numero vs. versus, from the Latin 631 a.i. affinely independent (§2.4.2.3) c.i. conically independent (§2.10) l.i. linearly independent w.r.t with respect to a.k.a also known as re real part im imaginary part ı or √ 1 − subset, superset ⊆ ⊇ proper subset, proper superset ⊂ ⊃ intersection, union ∩ ∪ membership, element belongs to, or element is a member of ∈ membership, contains as in y ( contains element y) ∋ C ∋ C Ä such that there exists ∃ ∴ therefore for all, or over all ∀ & (ampersand) and & (ampersand italic) and proportional to ∝ infinity ∞ equivalent to ≡ , defined equal to, equal by definition approximately equal to ≈ isomorphic to or with ≃ ∼= congruent to or with x Hadamard quotient as in, for x,y Rn, , [ x /y , i=1 ...n ] Rn ∈ y i i ∈ n § Hadamard product of matrices: x y , [ x y , i=1 ...n ] R (§D.1.2.2, A.1.1) ◦ ◦ i i ∈ § Kronecker product of matrices (§D.1.2.1, A.1.1) ⊗ vector sum of sets = where every element x has unique expression ⊕ x = y + z where yX Yand ⊕ Z z ; [349, p.19] then∈X summands are algebraic complements. = ∈Y =∈ Z+ . Now assume and are nontrivial X Y⊕Z⇒X Y Z Y Z § subspaces. = + = = 0 [350, §1.2] [123, 5.8]. Each element fromX a vectorY Z sum ⇒X (+) ofY⊕Z subspaces ⇔ Y∩Z has unique expression ( ) when a basis from each subspace is linearly independent of bases from all the other⊕ subspaces. 632 APPENDIX F. NOTATION, DEFINITIONS, GLOSSARY likewise, unique vector difference of sets ⊖ ⊞ orthogonal vector sum of sets = ⊞ where every element x has unique orthogonal expression x = y +Xz whereY Z y , z , and y ∈Xz . [372, p.51] ∈Y ∈ Z ⊥ = ⊞ = + . If ⊥ then = ⊞ = . [123, §5.8] X Y Z⇒X Y Z Z⊆Y X Y Z ⇔X Y ⊕ Z If = ⊥ then summands are orthogonal complements. Z Y plus or minus or plus and minus ± as in means logical not , or relative complement of set ; id est, \ = \Ax / ; e.g, , x A x / A \A { ∈A} B\A { ∈ B | ∈A} ≡ B∩\A and sufficient and necessary, implies and is implied by; e.g, ⇒ ⇐ A is sufficient: A B , A is necessary: A B , A B A ⇒B , A B A ⇐ B , if A⇒then⇔B \, ⇐ \ ⇐ ⇔if B \ then⇒ \A , A only if B , B only if A . if and only if (iff) or corresponds with or necessary and sufficient or logical ⇔ equivalence is plural are, as in A is B means A B ; conventional usage of English language imposed by logicians ⇒ ; and : insufficient and unnecessary, does not imply and is not implied by; e.g, A is insufficient: A ; B , A is unnecessary: A : B , A ; B A : B , A : B A ; B . ⇔ \ \ ⇔ \ \ is replaced with; substitution, assignment ← goes to, or approaches, or maps to → t 0+ t goes to 0 from above; meaning, from the positive [230, p.2] → . .. as in 1 1 means ones in a row or ··· ··· [ s1 sN ] means continuation; a matrix whose columns are si for i=1 ...N or··· as in n(n 1)(n 2) 1 means continuation of a product − − ··· . as in i=1 ...N meaning, i is a sequence of successive integers beginning with 1 and ending with N ; id est, 1 ...N = 1:N : as in f : Rn Rm meaning f is a mapping, or sequence→ of successive integers specified by bounds as in i:j = i...j (if j < i then sequence is descending) f real function or multidimensional function a.k.a operator f : meaning f is a mapping from ambient space to ambient , not necessarily M → R denoting either domain or range M R as in f(x) x means with the condition(s) or such that or evaluated for, or | as in f(x)| x∈C means evaluated for each and every x belonging to set { | ∈ C} C g expression g evaluated at xp |xp parallel k A,B as in, for example, A,B SN means A SN and B SN ∈ ∈ ∈ 633 (a, b) open interval between a and b in R or variable pair perhaps of disparate dimension [a, b ] closed interval or line segment between a and b in R ( ) hierarchal, parenthetical, optional 1 , k = 0 k ( n+k 1)! 1 − − , k > 0 − k!( n 1)! n k − −( k 1)! n 1 − − , k n n < 0 [260] 2 , − (n k)!( n 1)! binomial coefficient on Z k − − − − ≤ µ ¶ 0 , n < k < 0 0 , k < 0 or k > n n! n 0 k!(n k)! , otherwise ≥ − ¾ ! factorial; id est, for integer n>0 , n! , n(n 1)(n 2) 1 , ( n)!, , 0!,1 − − ··· − ∞ curly braces denote a set or list; e.g, Xa a 0 the set comprising Xa evaluated { } for each and every a 0 where membership{ | ofºa}to some space is implicit, a union; or 0, 1 n representsº a binary vector of dimension n { } angle brackets denote vector inner-product (35) (40) h i [ ] matrix or vector, or quote insertion, or citation th [dij ] matrix whose ij entry is dij th [xi] vector whose i entry is xi xp particular value of x x0 particular instance of x , or initial value of a sequence xi x first entry of vector x , or first element of a list x 1 { i} xε extreme point 1 x+ vector x whose negative entries are replaced with 0 : x+ = 2 (x + x ) (546) nonnegative part of x or clipped vector x | | 1 x x , (x x ) : nonpositive part of x = x+ + x − − 2 − | | − xˇ known data x⋆ optimal value of variable x . optimal feasible ⇒ x∗ complex conjugate or dual variable or extreme direction of dual cone f ∗ convex conjugate function f ∗(s) = sup s,x f(x) x dom f {h i − | ∈ } P x or Px projection of point x on set , P is operator or idempotent matrix C C P x projection of point x on set or on range of implicit vector k Ck δ(A) (a.k.a diag(A) , §A.1) vector made from main diagonal of A if A is a matrix; otherwise, diagonal matrix made from vector A δ2(A) δ(δ(A)). For vector or diagonal matrix Λ , δ2(Λ) = Λ ≡ δ(A)2 = δ(A)δ(A) where A is a vector 634 APPENDIX F.