definition 5.47 Inverse of a linear transformation

open in the book · parts/02-mathematical-methods/03-linear-algebra-representations.tex:2239 · p. 123

ground object -- no derivation owed

Rests on

depends_on definition 5.37 Linear transformation ¶
- 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 4.32 Field ¶
    - depends_on definition 4.11 Inverse element ¶
      - depends_on definition 4.9 Neutral element ¶
    - depends_on definition 4.20 Monoid ¶
      - depends_on definition 4.9 Neutral element ¶ ↺
      - depends_on definition 4.18 Semigroup ¶
    - depends_on definition 4.30 Ring ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶ ↺
      - depends_on definition 4.28 Distributivity ¶
      - depends_on definition 4.21 Group ¶
      - 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 12.49 Resolvent set; spectrum ¶
- depends_on definition 12.50 Point, continuous and residual spectrum ¶
  - depends_on definition 12.105 Generalized eigenvector ¶
    - depends_on proposition 12.106 The plane wave is a generalized momentum eigenvector ¶
      - depends_on example A.283 Momentum on the line ¶
    - depends_on theorem A.279 Gelfand–Maurin ¶
      - depends_on example A.283 Momentum on the line ¶ ↺
    - depends_on theorem 12.107 Nuclear spectral theorem ¶
  - 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 definition A.248 Cyclic vector and cyclic subspace ¶
      - depends_on proposition A.246 Bounded Borel functional calculus ¶
      - depends_on proposition A.245 The measures $\mu_{x,y}$ ¶
- 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 A.262 Spectral theorem for an unbounded self-adjoint operator ¶
      - depends_on proposition A.280 Direct-integral form of the spectral theorem ¶
  - 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 12.55 The spectrum of a self-adjoint operator is real ¶
  - depends_on theorem 12.54 The spectrum is compact and non-empty ¶ ↺
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 corollary 5.99 Semisimple and nilpotent parts ¶
  - 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 proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
    - depends_on lemma 44.7 Variation of inverse and determinant ¶
      - depends_on proposition 44.17 Stress–energy of the electromagnetic field ¶
      - depends_on theorem 44.9 Variation of the Einstein–Hilbert action ¶
    - 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 lemma A.742 Isotropic representation ¶
      - depends_on lemma A.671 The three products are independent ¶
      - depends_on lemma 30.26 Isotropic Cartesian tensors of rank four ¶
      - depends_on remark A.677 The lower ranks, and why an isotropic solid is not piezoelectric ¶
      - depends_on remark 30.27 Two results this chapter borrows from Part II ¶
      - depends_on remark 13.7 What the classification is used for ¶
  - 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 corollary 5.91 The operator norm of a real array ¶
      - depends_on lemma A.610 Block positivity ¶
      - depends_on lemma A.601 Several variables, complex symmetric matrix ¶
      - depends_on lemma A.600 The real part of an inverse ¶
      - depends_on proposition 5.90 Polar decomposition ¶ ↺
      - depends_on proposition 5.94 Principal axes of a real quadratic form ¶
      - 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 theorem 9.26 Structure of the solutions ¶
      - depends_on corollary 9.27 Linear stability ¶
- depends_on proposition 5.51 prop:lin-matrix-inverse ¶
- depends_on proposition 5.136 $\Sp(2n,\R)$ is a subgroup of $\GL(2n,\R)$ ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Linear transformation	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2249
`depends_on`	←	Resolvent set; spectrum	declared	parts/02-mathematical-methods/10-hilbert-spaces.tex:1309
`depends_on`	←	Existence, uniqueness, and linearity of the inverse	declared	parts/02-mathematical-methods/03-linear-algebra-representations.tex:2262