proposition 5.90 Polar decomposition

Rests on

• depends_on proposition 5.49 Injective, surjective, invertible ¶
  • depends_on definition 5.39 Kernel, image, nullity, rank ¶
    • 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 4.32 Field ¶
        • depends_on definition 4.31 Module ¶
    • depends_on definition 5.7 Vector subspace ¶
      • depends_on definition 5.5 Linear combination ¶
        • depends_on definition 4.33 Vector space ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-12 ¶
  • depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
    • depends_on definition 5.47 Inverse of a linear transformation ¶
      • depends_on definition 5.37 Linear transformation ¶ ↺
    • depends_on definition 5.37 Linear transformation ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-17 ¶
  • depends_on theorem 5.40 Rank–nullity ¶
    • depends_on definition 5.15 Basis ¶
      • depends_on definition 5.12 Subspace generated by a set of vectors ¶
        • depends_on definition 5.5 Linear combination ¶ ↺
        • depends_on definition 5.7 Vector subspace ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-3 ¶
      • depends_on definition 5.14 Linear independence ¶
        • depends_on definition 5.5 Linear combination ¶ ↺
    • depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
    • depends_on lemma 5.38 Exchange and completion ¶
      • depends_on definition 5.15 Basis ¶ ↺
      • depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
      • depends_on definition 5.14 Linear independence ¶ ↺
      • proves proof ch:03-linear-algebra-representations@proof-11 ¶
    • proves proof ch:03-linear-algebra-representations@proof-13 ¶
  • proves proof ch:03-linear-algebra-representations@proof-18 ¶
• depends_on theorem 5.84 Spectral theorem for a real symmetric operator ¶
  • depends_on proposition 5.28 Gram–Schmidt ¶
    • depends_on definition 5.14 Linear independence ¶ ↺
    • depends_on definition 5.18 Inner product ¶
      • depends_on definition 4.33 Vector space ¶ ↺
    • depends_on definition 5.27 Orthonormal basis ¶
      • depends_on definition 5.26 Orthogonal basis ¶
        • depends_on definition 5.15 Basis ¶ ↺
        • depends_on definition 5.24 Orthogonal vectors ¶
      • depends_on definition 5.25 Unit vector ¶
        • depends_on definition 5.19 Norm ¶
    • depends_on equation 5.44 eq:lin-norm-assoc ¶
    • proves proof ch:03-linear-algebra-representations@proof-5 ¶
  • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
    • depends_on definition 5.70 Characteristic polynomial ¶
      • depends_on definition 5.56 Endomorphism ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
      • depends_on equation 5.19 eq:lin-leibniz-det ¶
      • depends_on equation 5.74 eq:lin-matrix-rep ¶
      • proves proof ch:03-linear-algebra-representations@proof-25 ¶
    • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      • depends_on definition 5.56 Endomorphism ¶ ↺
      • depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
    • depends_on proposition 5.49 Injective, surjective, invertible ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-26 ¶
  • depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    • depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      • depends_on theorem 8.19 Fundamental theorem of algebra ¶
        • depends_on theorem 8.18 Liouville ¶
        • proves proof ch:06-complex-analysis@proof-13 ¶
      • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶ ↺
      • proves proof ch:03-linear-algebra-representations@proof-27 ¶
    • depends_on definition 5.41 Adjoint ¶
      • depends_on definition 5.18 Inner product ¶ ↺
      • depends_on definition 5.37 Linear transformation ¶ ↺
    • depends_on theorem 5.40 Rank–nullity ¶ ↺
    • proves proof ch:03-linear-algebra-representations@proof-31 ¶
  • proves proof ch:03-linear-algebra-representations@proof-34 ¶
• proves proof ch:03-linear-algebra-representations@proof-38 ¶

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