Math 2.0

Geometry Metric and reflection geometry Theorem

Cartan–Dieudonné theorem

Builds on ReflectionOrthogonal matrix

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.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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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 page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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

Further reading

  1. Web https://en.wikipedia.org/wiki/Cartan–Dieudonné_theorem, Formal statement
About this page
Tag0VMWGYY35Q
URIm2://0VMWGYY35Q
Kindstatement, theorem
WritingW1 stated
MSC51F15, 15B10
StatusThis copy of the site has no build data. Run m2 status 0VMWGYY35Q.
UsesReflection, Orthogonal matrix
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