Rigidity in normed spaces Rigidity in ( M n , � · � ) Infinitesimal rigidity for unitarily invariant matrix norms Derek Kitson Lancaster University IWOTA, TU Chemnitz 17th August 2017 Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Statement of the problem ( X, � · � ) is a finite dimensional real normed linear space. Problem: Given a framework ( G, p ) in X determine whether ( G, p ) is infinitesimally rigid (or isostatic) in ( X, � · � ) . Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Statement of the problem ( X, � · � ) is a finite dimensional real normed linear space. Problem: Given a framework ( G, p ) in X determine whether ( G, p ) is infinitesimally rigid (or isostatic) in ( X, � · � ) . Questions to consider ◮ Which motions are considered trivial? Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Statement of the problem ( X, � · � ) is a finite dimensional real normed linear space. Problem: Given a framework ( G, p ) in X determine whether ( G, p ) is infinitesimally rigid (or isostatic) in ( X, � · � ) . Questions to consider ◮ Which motions are considered trivial? ◮ What form does the infinitesimal flex condition take? Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Statement of the problem ( X, � · � ) is a finite dimensional real normed linear space. Problem: Given a framework ( G, p ) in X determine whether ( G, p ) is infinitesimally rigid (or isostatic) in ( X, � · � ) . Questions to consider ◮ Which motions are considered trivial? ◮ What form does the infinitesimal flex condition take? ◮ Is infinitesimal rigidity a generic property? Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) What is known Euclidean norm ◮ discrete geometry, combinatorics, semidefinite programming, operator theory... Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) What is known Euclidean norm ◮ discrete geometry, combinatorics, semidefinite programming, operator theory... General norms ◮ flex condition, rigidity matrix, symmetry Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) What is known Euclidean norm ◮ discrete geometry, combinatorics, semidefinite programming, operator theory... General norms ◮ flex condition, rigidity matrix, symmetry ℓ p norms, p / ∈ { 1 , 2 , ∞} ◮ Laman-type theorems Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) What is known Euclidean norm ◮ discrete geometry, combinatorics, semidefinite programming, operator theory... General norms ◮ flex condition, rigidity matrix, symmetry ℓ p norms, p / ∈ { 1 , 2 , ∞} ◮ Laman-type theorems Polyhedral norms ◮ edge-colouring techniques Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) What is known Euclidean norm ◮ discrete geometry, combinatorics, semidefinite programming, operator theory... General norms ◮ flex condition, rigidity matrix, symmetry ℓ p norms, p / ∈ { 1 , 2 , ∞} ◮ Laman-type theorems Polyhedral norms ◮ edge-colouring techniques Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Unitarily invariant norms Let M n denote the vector space of n × n matrices (over R or C ). A norm on M n is unitarily invariant if � a � = � uav � for all a ∈ M n and all unitary matrices u, v ∈ M n . Theorem (von Neumann, 1937) A matrix norm is unitarily invariant if and only if it is obtained by applying a symmetric norm to the vector of singular values of a matrix. Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Examples of unitarily invariant norms The Schatten p -norms on M n are defined by, � n � 1 p � σ p � a � c p = , 1 ≤ p < ∞ , i i =1 � a � c ∞ = max σ i , i where σ i are the singular values of a . ◮ c 1 = trace norm ◮ c 2 = Frobenius norm (= Euclidean norm of matrix entries) ◮ c ∞ = spectral norm (= operator norm on Euclidean space) Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Rigid motions A rigid motion of a normed space ( X, � · � ) is a collection of continuous paths α = { α x : [ − 1 , 1] → X } x ∈ X , with the following properties: (a) α x (0) = x for all x ∈ X ; (b) α x ( t ) is differentiable at t = 0 for all x ∈ X ; and (c) � α x ( t ) − α y ( t ) � = � x − y � for all x, y ∈ X and for all t ∈ [ − 1 , 1] . We write R ( X, � · � ) for the set of all rigid motions of ( X, � · � ) . Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Rigid motions Lemma Let ( X, � · � ) be a normed space and let α ∈ R ( X, � · � ) . Then, (i) for each t ∈ [ − 1 , 1] there exists a real-linear isometry A t : X → X and a vector c ( t ) ∈ X such that α x ( t ) = A t ( x ) + c ( t ) , ∀ x ∈ X. (ii) the map c : [ − 1 , 1] → X is continuous on [ − 1 , 1] and differentiable at t = 0 , (iii) for every x ∈ X , the map A ∗ ( x ) : [ − 1 , 1] → X , t �→ A t ( x ) , is continuous on [ − 1 , 1] and differentiable at t = 0 , and, (iv) A 0 = I and c (0) = 0 . Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Rigid motions Proposition For any α ∈ R ( M n , � · � ) , there is a neighbourhood T of 0 in [ − 1 , 1] , and matrices u t , w t ∈ U n and c ( t ) ∈ M n for each t ∈ T , so that (i) α x ( t ) = u t xw t + c ( t ) , ∀ x ∈ M n , t ∈ T ; (ii) c (0) = 0 and u 0 = w 0 = I ; (iii) the maps t �→ c ( t ) and t �→ u t xw t are both differentiable at t = 0 , for any x ∈ M n ; and (iv) the maps t �→ u t and t �→ w t are continuous at t = 0 . Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Infinitesimal rigid motions A vector field η : X → X of the form η ( x ) = α ′ x (0) where α ∈ R ( X, � · � ) is referred to as an infinitesimal rigid motion of ( X, � · � ) . Lemma Let ( X, � · � ) be a normed space and let η ∈ T ( X, � · � ) . Then η is an affine map. Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Infinitesimal rigid motions Theorem If η ∈ T ( M n , � · � ) , then there exist unique matrices a, b, c ∈ M n with a ∈ Skew 0 n , b ∈ Skew n and c ∈ M n so that η ( x ) = ax + xb + c, ∀ x ∈ M n . Define Ψ : T ( M n , � · � ) → Skew 0 n ⊕ Skew n ⊕ M n by setting Ψ X ( η ) = ( a, b, c ) if and only if η ( x ) = ax + xb + c for all x ∈ X . Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Infinitesimal rigid motions Lemma Ψ is a linear isomorphism. Proof. Let ( a, b, c ) be in the codomain of Ψ , and for each x ∈ M n define α x ( t ) = e ta xe tb + tc. α x : [ − 1 , 1] → M n , Since a and b are skew-hermitian, e ta and e tb are unitary for every t ∈ R , so { α x } x ∈M n is a rigid motion. The induced infinitesimal rigid motion is the vector field η : M n → M n , x �→ ax + xb + c. Thus Ψ( η ) = ( a, b, c ) and so Ψ is surjective. Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Infinitesimal rigid motions Proposition 2 n 2 − n � if K = R , dim T ( M n , � · � ) = 4 n 2 − 1 if K = C . Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
Rigidity in normed spaces Rigidity in ( M n , � · � ) Support functionals A support functional for a unit vector x 0 ∈ X is a linear functional f : X → R with � f � := sup {| f ( x ) | : x ∈ X, � x � = 1 } ≤ 1 , and f ( x 0 ) = 1 . Derek Kitson Lancaster University Infinitesimal rigidity for unitarily invariant matrix norms
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