Math 2.0

statement theorem

Cartan–Dieudonné theorem

Statement

Let VV be a real inner product space of finite dimension nn. Then each linear isometry T ⁣:V→VT \colon V \to V is a composition of at most nn reflections in hyperplanes of VV.

Remarks

The identity map is the composition of zero reflections. In R2\R^2, each rotation is the composition of two reflections in lines through the origin. The theorem holds more generally for nondegenerate symmetric bilinear forms on spaces over fields of characteristic not 22.

Explanations

L1 human

Every rigid linear motion of nn-dimensional space is a sequence of at most nn mirror reflections. So reflections are the building blocks of all isometries. In the plane, a rotation by θ\theta is the product of two reflections in lines at the angle θ/2\theta/2.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionTranspose, definition, no evidence levelTransposedefinitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionA space is the direct sum of a subspace and its orthogonal complement, statement, no evidence levelA space is the direct…statementProjection, definition, no evidence levelProjectiondefinitionOrthogonal projection, definition, no evidence levelOrthogonal projectiondefinitionReflection, definition, no evidence levelReflectiondefinitionCartan–Dieudonné theorem, statement, no evidence levelCartan–Dieudonné theo…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Transpose definition
  5. Inner product definition
  6. Linear map definition
  7. Linear subspace definition
  8. Orthogonal matrix definition
  9. Orthogonal vectors definition
  10. Endomorphism definition
  11. Sum of subspaces definition
  12. Orthogonal complement definition
  13. Direct sum of subspaces definition
  14. A space is the direct sum of a subspace and its orthogonal complement statement
  15. Projection definition
  16. Orthogonal projection definition
  17. Reflection definition
  18. Cartan–Dieudonné theorem statement

Sources

  1. Web https://en.wikipedia.org/wiki/Cartan–Dieudonné_theorem, Formal statement

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