definition 5.37 Linear transformation

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:1702 · p. 116

ground object -- no derivation owed

Rests on

depends_on definition 4.33 Vector space ¶
- depends_on definition 4.8 Commutativity; abelian structure ¶
  - depends_on definition 4.4 Internal binary operation; magma ¶
    - depends_on definition 4.3 Algebraic structure ¶
      - depends_on definition 4.2 Binary operation ¶
    - depends_on definition 4.2 Binary operation ¶ ↺
- depends_on definition 4.32 Field ¶
  - depends_on definition 4.11 Inverse element ¶
    - depends_on definition 4.9 Neutral element ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
  - depends_on definition 4.20 Monoid ¶
    - depends_on definition 4.9 Neutral element ¶ ↺
    - depends_on definition 4.18 Semigroup ¶
      - depends_on definition 4.6 Associativity ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶ ↺
  - depends_on definition 4.30 Ring ¶
    - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
    - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.3 Algebraic structure ¶ ↺
    - depends_on definition 4.21 Group ¶
      - depends_on definition 4.11 Inverse element ¶ ↺
      - depends_on definition 4.20 Monoid ¶ ↺
    - depends_on definition 4.18 Semigroup ¶ ↺
- depends_on definition 4.31 Module ¶
  - depends_on definition 4.29 Action ¶
    - depends_on definition 4.9 Neutral element ¶ ↺
  - depends_on definition 4.28 Distributivity ¶ ↺
  - depends_on definition 4.21 Group ¶ ↺
  - depends_on definition 4.30 Ring ¶ ↺

Supports

depends_on definition 7.99 Differentiability at a point ¶
- depends_on definition 7.102 Differential of a function ¶
  - depends_on proposition 7.104 Chain rule in several variables ¶
    - depends_on corollary 7.113 Inverse function theorem ¶
      - depends_on definition A.503 The substitution property ¶
      - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
      - depends_on lemma A.510 Every diffeomorphism factorises locally ¶
      - depends_on remark 7.114 Where the proof is, and why ¶
      - depends_on theorem 7.129 Change of variables in a multiple integral ¶
      - depends_on theorem 7.115 Constant rank ¶
    - depends_on corollary 7.110 Variable limits of integration ¶
    - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on lemma A.549 The action is transitive ¶
      - depends_on lemma A.300 The boundary is well defined, and is a manifold ¶
      - depends_on proposition A.535 Existence of a slice ¶
      - depends_on proposition 22.32 Duality of the two brackets ¶
      - depends_on proposition 22.3 Invertibility of the Legendre map ¶
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on remark 22.19 Which form exists ¶
      - depends_on theorem 13.63 Constant rank theorem ¶
    - depends_on definition 30.4 Material and spatial descriptions ¶
    - depends_on definition 31.4 Material derivative ¶
      - depends_on definition 31.6 Streamline, pathline, streakline ¶
      - depends_on lemma 31.11 Transport theorem for a material volume ¶
    - depends_on definition 22.26 Lagrange bracket ¶
    - depends_on definition 10.18 Complete integral; envelope ¶
      - depends_on proposition 10.19 An envelope of solutions is a solution ¶
    - depends_on lemma A.510 Every diffeomorphism factorises locally ¶ ↺
    - depends_on lemma A.504 Transitivity ¶
    - depends_on lemma A.306 The standard smooth bump ¶
      - depends_on lemma A.307 Partition of unity on a compact manifold ¶
    - depends_on proposition 7.136 Properties of conservative fields ¶
      - depends_on example 31.15 The stream function of a plane flow ¶
      - depends_on proposition 31.28 Potential flow reduces to Laplace's equation ¶
      - depends_on theorem 16.63 Hilbert's invariant integral ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
    - depends_on proposition 7.120 The envelope touches every member it meets ¶
    - … 9 more
  - depends_on remark 7.103 rem:ana-differential-scalar ¶
- depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
- depends_on proposition 7.104 Chain rule in several variables ¶ ↺
- depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
  - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶ ↺
  - depends_on remark 7.101 Partial derivatives alone do not suffice ¶
  - depends_on theorem 7.112 Implicit function theorem ¶
    - depends_on corollary 7.113 Inverse function theorem ¶ ↺
    - depends_on proposition 7.120 The envelope touches every member it meets ¶ ↺
    - depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶
      - depends_on remark 7.118 The multiplier rule and its functional counterpart ¶
    - depends_on remark 7.114 Where the proof is, and why ¶ ↺
  - depends_on theorem A.286 Implicit function theorem ¶
    - depends_on corollary A.292 Inverse function theorem ¶ ↺
    - depends_on corollary A.293 Solving one scalar equation for one coordinate ¶
      - depends_on example A.294 The sphere, made explicit ¶
    - depends_on lemma A.650 Flattening the constraints ¶
      - depends_on lemma A.651 Acyclicity of $\delta$ ¶
    - depends_on remark A.296 Both hypotheses are needed, and the conclusion is local ¶
    - depends_on remark 23.3 What Part II owes this chapter ¶
    - depends_on theorem 23.6 Jacobi ¶
      - depends_on example 23.25 Central force in plane polar coordinates ¶
      - depends_on example 23.22 Free particle ¶
      - depends_on example 23.23 Uniform gravity, and the parabola ¶
      - depends_on proposition 23.7 The principal function is the action ¶
    - depends_on theorem 13.63 Constant rank theorem ¶ ↺
    - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
      - depends_on corollary 13.64 The image of a constant-rank map, locally ¶
- depends_on theorem 7.112 Implicit function theorem ¶ ↺
- depends_on theorem A.286 Implicit function theorem ¶ ↺
depends_on definition 12.35 Bounded operator; operator norm ¶
- depends_on definition 12.45 Continuous linear functional; the dual ¶
  - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶
    - depends_on definition 12.103 Gelfand triple ¶
      - depends_on definition A.278 Countably Hilbert nuclear space ¶
      - depends_on definition 12.105 Generalized eigenvector ¶
      - depends_on example 12.104 The Schwartz triple ¶
      - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on theorem 12.107 Nuclear spectral theorem ¶
  - depends_on definition 12.103 Gelfand triple ¶ ↺
  - depends_on theorem 12.46 Riesz representation ¶
    - depends_on corollary 12.47 $\mathcal{H}$ is its own dual, antilinearly ¶ ↺
    - depends_on lemma A.247 Integration against a projection-valued measure ¶
    - depends_on proposition A.246 Bounded Borel functional calculus ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶
      - depends_on proposition A.261 Spectral theorem for a unitary operator ¶
    - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
      - depends_on lemma A.249 The cyclic case ¶
      - depends_on proposition A.246 Bounded Borel functional calculus ¶ ↺
    - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      - depends_on definition 12.41 The operator classes ¶
      - depends_on definition 12.71 Adjoint of a densely defined operator ¶
      - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶
- depends_on definition 12.88 Reducing subspace ¶
  - depends_on definition 12.90 Self-adjoint family; commutant; irreducibility ¶
    - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶
      - depends_on lemma A.590 The Gram matrix is universal ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶
    - depends_on theorem A.579 Stone–von Neumann ¶
    - depends_on theorem 12.91 Schur's lemma, commutant form ¶
      - depends_on proposition A.591 Any two irreducible Weyl systems are equivalent ¶ ↺
      - depends_on theorem 12.114 Stone–von Neumann ¶
    - depends_on theorem 12.114 Stone–von Neumann ¶ ↺
  - depends_on proposition 12.89 Reduction is commutation ¶
    - depends_on proposition A.589 Cyclic subspaces and the rank of the average ¶ ↺
    - depends_on theorem 12.91 Schur's lemma, commutant form ¶ ↺
- depends_on definition 12.49 Resolvent set; spectrum ¶
  - depends_on definition 12.50 Point, continuous and residual spectrum ¶
    - depends_on definition 12.105 Generalized eigenvector ¶ ↺
    - depends_on example 12.56 Multiplication by the coordinate: spectrum without eigenvectors ¶
    - depends_on proposition 12.51 The three cases are exclusive and exhaustive ¶
  - depends_on lemma A.240 Spectral mapping for polynomials ¶
    - depends_on proposition A.241 The polynomial calculus is isometric ¶
      - depends_on proposition A.243 Continuous functional calculus ¶
  - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶
    - depends_on proposition A.241 The polynomial calculus is isometric ¶ ↺
  - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
    - depends_on lemma A.239 The norm of a self-adjoint operator lies in its spectrum ¶ ↺
    - depends_on proposition A.261 Spectral theorem for a unitary operator ¶ ↺
    - depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
      - depends_on theorem 12.54 The spectrum is compact and non-empty ¶
    - depends_on theorem 12.54 The spectrum is compact and non-empty ¶ ↺
- depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
  - depends_on lemma A.240 Spectral mapping for polynomials ¶ ↺
  - depends_on proposition 12.65 Exponential of a bounded self-adjoint operator ¶
    - depends_on theorem 12.66 Stone ¶
      - depends_on corollary 12.112 Commutator of the momentum with a function of the position ¶
      - depends_on definition 12.109 Weyl system ¶
      - depends_on proposition 12.67 The generator is symmetric, and generates the motion ¶
      - depends_on proposition 12.111 The Weyl relation is a covariance statement ¶
      - depends_on theorem 25.34 Stone–von Neumann ¶
  - depends_on proposition 12.61 Uniqueness of the continuous functional calculus ¶
    - depends_on proposition A.243 Continuous functional calculus ¶ ↺
  - depends_on proposition 12.39 Algebra of the adjoint; the $C^{\ast}$ identity ¶ ↺
  - depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶ ↺
  - depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
  - depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶
    - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶
    - depends_on corollary 25.30 The relation cannot be realized by matrices ¶
    - depends_on definition 12.109 Weyl system ¶ ↺
- depends_on proposition 12.36 Boundedness is continuity ¶
  - depends_on theorem 12.74 Hellinger–Toeplitz ¶
    - depends_on corollary 12.76 Position and momentum are unbounded, and cannot be everywhere defined ¶ ↺
- depends_on theorem 12.75 The canonical commutation relation admits no bounded solution ¶ ↺
depends_on definition 12.69 Operator with a domain ¶
- depends_on definition 12.70 Graph; closed and closable operators ¶
  - depends_on proposition 12.73 The adjoint is always closed ¶
    - depends_on definition 12.78 Essential self-adjointness ¶
      - depends_on theorem A.266 von Neumann ¶
      - depends_on theorem 12.80 von Neumann's criterion ¶
    - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶
      - depends_on proposition A.262 Spectral theorem for an unbounded self-adjoint operator ¶ ↺
- depends_on definition 12.72 Symmetric; self-adjoint ¶
  - depends_on definition 12.78 Essential self-adjointness ¶ ↺
  - depends_on lemma A.260 Cayley transform of a self-adjoint operator ¶ ↺
  - depends_on lemma A.267 Isometry of $A\pm\ii\mu$, and closed range ¶
    - depends_on definition A.269 Cayley transform ¶
      - depends_on proposition A.270 Properties of the transform ¶
    - depends_on lemma A.268 The indices do not depend on $\mu$ ¶
    - depends_on proposition A.270 Properties of the transform ¶ ↺
    - depends_on proposition A.273 Self-adjoint means unitary ¶
  - depends_on lemma A.271 Injectivity of $\identity-V$ for any isometric extension ¶
    - depends_on lemma A.272 The operator attached to an isometry ¶
  - depends_on lemma A.272 The operator attached to an isometry ¶ ↺
  - depends_on proposition A.273 Self-adjoint means unitary ¶ ↺
  - depends_on theorem A.266 von Neumann ¶ ↺
  - depends_on theorem 12.74 Hellinger–Toeplitz ¶ ↺
  - depends_on theorem 12.80 von Neumann's criterion ¶ ↺
- depends_on definition 12.71 Adjoint of a densely defined operator ¶ ↺
depends_on definition 5.41 Adjoint ¶
- depends_on proposition 5.42 The adjoint exists, is unique, and is linear ¶
  - depends_on corollary 5.82 Spectral theorem for a normal operator ¶
- depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶ ↺
- depends_on theorem 5.43 The four fundamental subspaces ¶
- depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶
  - depends_on corollary 5.82 Spectral theorem for a normal operator ¶ ↺
- depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
  - depends_on definition 30.20 Pressure and deviatoric stress ¶
    - depends_on definition 30.31 The engineering moduli ¶
      - depends_on phenomenon 30.70 Hertzian contact ¶
      - depends_on phenomenon 30.60 Coulomb's law of torsion ¶
      - depends_on proposition 30.33 Conversion among the moduli ¶
      - depends_on proposition 30.51 Moment and curvature ¶
      - depends_on remark A.676 The third cubic constant is measured, and it is not small ¶
      - depends_on theorem A.695 Boussinesq's point-force solution, quoted ¶
    - depends_on definition 30.79 Maxwell viscoelasticity ¶
      - depends_on proposition 30.80 A Maxwell material is a solid or a fluid depending on how long you watch ¶
    - depends_on proposition 30.74 The maximum shear stress, and its indifference to pressure ¶
      - depends_on phenomenon 30.73 Yield and brittle fracture ¶
      - depends_on remark A.705 The dangerous stress is below the surface ¶
      - depends_on remark 30.75 Two criteria, one uncited ¶
    - depends_on remark 30.75 Two criteria, one uncited ¶ ↺
  - depends_on proposition 30.9 Principal strains ¶
    - depends_on example 30.10 Simple shear and pure shear are the same strain ¶
  - depends_on theorem 5.80 Simultaneous diagonalization of commuting self-adjoint operators ¶ ↺
  - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
    - depends_on corollary 5.91 The operator norm of a real array ¶
      - depends_on proposition 5.92 The eigenvalues do not control the norm ¶
    - depends_on lemma A.610 Block positivity ¶
      - depends_on proposition A.611 The constraint on $\Gamma$ ¶
    - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
      - depends_on proposition A.602 The Wigner function of a Gaussian is a positive Gaussian ¶
    - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶ ↺
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶ ↺
    - depends_on proposition 5.90 Polar decomposition ¶
    - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - depends_on proposition 5.95 The focal polar equation of a conic ¶
    - depends_on proposition 5.92 The eigenvalues do not control the norm ¶ ↺
    - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶
      - depends_on corollary 5.89 Rayleigh–Ritz ¶
      - depends_on proposition 5.88 Rayleigh quotient of a definite pencil ¶
  - depends_on theorem 28.33 Normal modes ¶
    - depends_on example 28.39 The linear triatomic molecule ¶
    - depends_on proposition 28.40 Modes and dispersion of the loaded string ¶
    - depends_on remark 28.34 Part II owes the simultaneous diagonalization of two quadratic forms ¶
  - depends_on theorem 29.16 Principal axes ¶
    - depends_on definition 29.19 Inertia ellipsoid ¶
      - depends_on theorem 29.29 Poinsot's construction ¶
    - depends_on definition 29.35 The Lagrange top ¶
    - depends_on definition 29.18 Classification of tops ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶
    - depends_on phenomenon 29.31 The intermediate-axis instability ¶
      - depends_on remark 29.32 What the linearization is entitled to conclude ¶
    - depends_on proposition 29.33 Dissipation drives a free body to its largest moment ¶
    - depends_on proposition 29.17 Triangle inequalities ¶
      - depends_on proposition 29.22 Perpendicular-axis theorem ¶
    - depends_on theorem 29.25 Euler's equations ¶
      - depends_on phenomenon 29.46 A fast spinning top does not fall ¶
      - depends_on phenomenon 29.31 The intermediate-axis instability ¶ ↺
      - depends_on proposition 29.26 The two integrals of torque-free motion ¶
      - depends_on proposition 29.28 Free rotation of a symmetric body ¶ ↺
      - depends_on proposition 24.41 The free rigid body is a Lie–Poisson system ¶
depends_on definition 5.110 Bilinear map ¶
- depends_on definition 12.94 Tensor product of Hilbert spaces ¶
  - depends_on definition 12.99 Product and entangled vectors ¶
    - depends_on example 12.100 Entangled vectors exist ¶
  - depends_on example 12.98 Two particles in three-dimensional space ¶
  - depends_on proposition 12.95 The tensor inner product is well defined and positive definite ¶
    - depends_on example 12.100 Entangled vectors exist ¶ ↺
    - depends_on proposition 12.96 Operators on a tensor product ¶
- depends_on definition 5.113 Non-degenerate form ¶
  - depends_on definition 5.117 Signature ¶
    - depends_on proposition 5.94 Principal axes of a real quadratic form ¶ ↺
  - depends_on definition 5.135 The symplectic group ¶
    - depends_on definition 22.40 Symplectic matrix ¶
    - depends_on definition 24.4 Symplectic group ¶
      - depends_on proposition 24.5 Properties of the symplectic group ¶
      - depends_on proposition 24.30 Linear non-squeezing ¶
    - depends_on lemma 22.20 The symplectic condition ¶
      - depends_on corollary 22.24 Invariance of the phase-space volume ¶
      - depends_on proposition 22.27 Properties of the Lagrange bracket ¶
      - depends_on proposition 22.30 Properties of the Poisson bracket ¶
      - depends_on remark 22.33 What the duality is good for ¶
      - depends_on remark 22.28 Where the content of the fundamental brackets lies ¶
      - depends_on theorem 22.22 Poincaré ¶
    - depends_on proposition 5.139 Dimension of the symplectic group ¶
      - depends_on proposition 24.5 Properties of the symplectic group ¶ ↺
      - depends_on remark 22.21 The count ¶
    - depends_on proposition 5.136 $\Sp(2n,\R)$ is a subgroup of $\GL(2n,\R)$ ¶
    - depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶
      - depends_on corollary 22.24 Invariance of the phase-space volume ¶ ↺
      - depends_on corollary 24.19 Canonical transformations preserve phase volume ¶
      - depends_on proposition 24.5 Properties of the symplectic group ¶ ↺
  - depends_on proposition 5.114 prop:lin-bilinear-dual ¶
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    - depends_on definition 5.135 The symplectic group ¶ ↺
    - depends_on definition 24.2 Symplectic vector space ¶
      - depends_on definition 24.8 Symplectic manifold ¶
      - depends_on lemma A.564 Dimension and double orthogonal ¶
      - depends_on proposition 24.3 Even dimension and the canonical basis ¶
    - depends_on proposition 26.19 Second-class constraints come in pairs ¶
      - depends_on definition 26.20 Dirac bracket ¶
      - depends_on proposition 26.24 The Faddeev–Popov determinant ¶
      - depends_on proposition 26.25 The Liouville measure of a second-class surface ¶
      - depends_on theorem 26.17 Counting the physical degrees of freedom ¶
      - depends_on theorem 26.21 Properties of the Dirac bracket ¶
    - depends_on proposition 24.3 Even dimension and the canonical basis ¶ ↺
    - depends_on theorem A.68 Darboux ¶
      - depends_on corollary A.69 No local invariants ¶
      - depends_on example A.80 The phase space of one particle in space ¶
      - depends_on theorem 24.12 Darboux ¶
    - depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶ ↺
- depends_on definition 5.111 Symmetric and antisymmetric forms ¶
  - depends_on definition 28.32 The two quadratic forms ¶
  - depends_on proposition 5.112 Symmetric–antisymmetric splitting ¶
  - depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
  - depends_on theorem 5.116 Sylvester's law of inertia ¶
    - depends_on definition 5.117 Signature ¶ ↺
    - depends_on definition 10.6 Type of a second-order operator ¶
      - depends_on definition 10.79 Parabolic boundary ¶
      - depends_on example 10.9 ex:pde-three-types ¶
      - depends_on proposition 10.25 The backward heat problem is ill posed ¶
      - depends_on theorem A.83 Local canonical form for an elliptic operator with Hölder coefficients ¶
      - depends_on theorem 10.8 The type is a coordinate invariant ¶
    - depends_on proposition 30.30 Stability bounds ¶
      - depends_on phenomenon 30.43 Two bulk wave speeds ¶
      - depends_on phenomenon 30.32 Poisson contraction ¶
      - depends_on proposition 30.34 Minimum of the potential energy ¶
      - depends_on theorem 30.42 Uniqueness in elastodynamics ¶
    - depends_on proposition 5.94 Principal axes of a real quadratic form ¶ ↺
    - depends_on remark 7.107 What the Hessian is for ¶
    - depends_on theorem 5.86 Simultaneous diagonalization of a definite pencil ¶ ↺
    - depends_on theorem 28.33 Normal modes ¶ ↺
    - depends_on theorem 10.8 The type is a coordinate invariant ¶ ↺
- depends_on lemma 5.137 The alternating top form is unique up to scale ¶
  - depends_on theorem 5.138 A symplectic transformation has determinant $+1$ ¶ ↺
- depends_on proposition 5.114 prop:lin-bilinear-dual ¶ ↺
- depends_on proposition 5.112 Symmetric–antisymmetric splitting ¶ ↺
- depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
- depends_on theorem 5.116 Sylvester's law of inertia ¶ ↺
depends_on definition 5.128 Derivation ¶
- depends_on remark 22.31 The bracket is a Lie algebra structure ¶
  - depends_on remark 22.39 Poisson's theorem does not manufacture new constants ¶
depends_on definition 5.56 Endomorphism ¶
- depends_on definition 5.57 Automorphism ¶
- depends_on definition 5.70 Characteristic polynomial ¶
  - depends_on definition 5.73 Algebraic and geometric multiplicity ¶
    - depends_on proposition 5.74 prop:lin-geometric-le-algebraic ¶
  - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on theorem 5.98 Primary decomposition ¶
      - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶ ↺
    - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶ ↺
    - depends_on theorem 9.26 Structure of the solutions ¶
      - depends_on corollary 9.27 Linear stability ¶
- depends_on definition 5.76 Diagonalizable operator ¶
  - depends_on corollary 5.99 Semisimple and nilpotent parts ¶
  - depends_on theorem 5.77 Criterion for diagonalizability ¶
    - depends_on proposition 32.39 Stability of a periodic orbit ¶
      - depends_on phenomenon 32.87 Universal period doubling ¶
      - depends_on proposition 32.84 Fixed points and the first period doubling ¶
    - depends_on theorem 32.13 Linear stability ¶
      - depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
- depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
  - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
    - depends_on corollary 9.27 Linear stability ¶ ↺
    - depends_on proposition 9.29 The planar classification ¶
  - depends_on definition 5.73 Algebraic and geometric multiplicity ¶ ↺
  - depends_on definition 9.30 Hyperbolic equilibrium ¶
    - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
  - depends_on proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶
    - depends_on theorem 5.77 Criterion for diagonalizability ¶ ↺
  - depends_on theorem 5.77 Criterion for diagonalizability ¶ ↺
  - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶ ↺
depends_on definition 5.46 Functional ¶
- depends_on definition 12.45 Continuous linear functional; the dual ¶ ↺
- depends_on definition 5.59 Dual space ¶
  - depends_on definition 5.66 Pushforward ¶
    - depends_on proposition 5.67 Adjointness of pullback and pushforward ¶
  - depends_on proposition 5.61 Annihilator of a subspace ¶
    - depends_on corollary 5.62 A functional that annihilates a set of constraints ¶
  - depends_on proposition 5.114 prop:lin-bilinear-dual ¶ ↺
  - depends_on proposition 5.63 prop:lin-dual-inner-product ¶
  - depends_on proposition 5.67 Adjointness of pullback and pushforward ¶ ↺
- depends_on definition 5.60 Functional specified on a basis ¶
depends_on definition 5.47 Inverse of a linear transformation ¶
- depends_on definition 12.49 Resolvent set; spectrum ¶ ↺
- depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
  - depends_on proposition 5.49 Injective, surjective, invertible ¶
    - depends_on lemma 5.97 Fitting splitting ¶
      - depends_on theorem 5.98 Primary decomposition ¶ ↺
    - depends_on proposition 5.31 Gram criterion ¶
    - depends_on proposition 5.2 Jacobi's formula, column form ¶
      - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
      - depends_on lemma 44.7 Variation of inverse and determinant ¶
      - depends_on proposition 7.111 Jacobi's formula, cofactor form ¶
    - depends_on proposition 5.90 Polar decomposition ¶ ↺
    - depends_on proposition 5.130 An orthogonal transformation is an isometry ¶
      - depends_on theorem 5.133 Isotropic Cartesian tensors of rank at most four ¶
    - depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶ ↺
  - depends_on proposition 5.51 prop:lin-matrix-inverse ¶
  - depends_on proposition 5.136 $\Sp(2n,\R)$ is a subgroup of $\GL(2n,\R)$ ¶ ↺
depends_on definition 5.53 Isomorphism ¶
- depends_on definition 5.57 Automorphism ¶ ↺
- depends_on proposition 5.104 Criterion for a direct sum ¶
- depends_on proposition 5.54 prop:lin-same-dim-isomorphic ¶
- depends_on theorem 5.160 Schur's second lemma ¶
  - depends_on lemma 25.39 The low-degree images are forced ¶
    - depends_on example 25.45 Weyl ordering collapses ¶
  - depends_on theorem 25.38 Groenewold–van Hove ¶
    - depends_on corollary 25.41 Quantization is not a functor ¶
      - depends_on definition 25.42 Ordering rule ¶
    - depends_on remark 24.57 The half-integer, and the honest status of the construction ¶
    - depends_on remark 24.55 What polarization costs ¶
depends_on definition 5.39 Kernel, image, nullity, rank ¶
- depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
- depends_on lemma 5.97 Fitting splitting ¶ ↺
- depends_on proposition 5.49 Injective, surjective, invertible ¶ ↺
- depends_on theorem 5.40 Rank–nullity ¶
  - depends_on corollary 5.62 A functional that annihilates a set of constraints ¶ ↺
  - depends_on lemma A.610 Block positivity ¶ ↺
  - depends_on lemma A.600 The real part of an inverse ¶ ↺
  - depends_on lemma A.564 Dimension and double orthogonal ¶ ↺
  - depends_on lemma 5.97 Fitting splitting ¶ ↺
  - depends_on proposition 5.49 Injective, surjective, invertible ¶ ↺
  - depends_on theorem 5.43 The four fundamental subspaces ¶ ↺
  - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶ ↺
depends_on definition 5.66 Pushforward ¶ ↺
… 6 more

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type	direction	node	provenance	where
`depends_on`	→	Vector space	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1712
`depends_on`	←	Differentiability at a point	declared	parts/02-mathematical-methods/05-real-analysis.tex:2953
`depends_on`	←	Bounded operator; operator norm	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:894
`depends_on`	←	Operator with a domain	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1944
`depends_on`	←	Adjoint	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1875
`depends_on`	←	Bilinear map	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:4744
`depends_on`	←	Derivation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5529
`depends_on`	←	Endomorphism	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2494
`depends_on`	←	Functional	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2131
`depends_on`	←	Inverse of a linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2249
`depends_on`	←	Isomorphism	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2406
`depends_on`	←	Kernel, image, nullity, rank	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:1808
`depends_on`	←	Pushforward	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2979
`depends_on`	←	A nilpotent map has nilpotent adjoint	declared	appendices/A-long-proofs.tex:16409
`depends_on`	←	Existence, uniqueness, and linearity of the inverse	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2262
`depends_on`	←	prop:lin-matrix-algebra	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:5360
`depends_on`	←	prop:lin-matrix-unique	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2073
`depends_on`	←	Jordan decomposition	declared	appendices/A-long-proofs.tex:16332
`depends_on`	←	Schur's first lemma	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:7001