Math 2.0

Operator theory Spectral theorems and functional calculus Theorem

Spectral theorem

Builds on Self-adjoint operatorOrthonormal basisEigenvector

Statement

Let VV be a finite-dimensional inner product space over ℝ\R or ℂ\C, and let T ⁣:V→VT \colon V \to V be a self-adjoint linear map. Then VV has an orthonormal basis that consists of eigenvectors of TT. The corresponding eigenvalues are real.

Remarks

In this basis, the matrix of TT is a real diagonal matrix. Equivalently, VV is the orthogonal direct sum of the eigenspaces of TT. Over ℂ\C, the same conclusion holds, with complex eigenvalues, for each normal operator. The matrix form is the spectral theorem for Hermitian matrices.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionEigenvector, definition, no evidence levelEigenvectordefinitionBasis, definition, no evidence levelBasisdefinitionOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionSpectral theorem, statement, no evidence levelSpectral theoremstatement
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. Vector space definition
  2. Inner product definition
  3. Linear combination definition
  4. Linear map definition
  5. Linear subspace definition
  6. Inner product space definition
  7. Orthogonal vectors definition
  8. Linear independence definition
  9. Endomorphism definition
  10. Linear span definition
  11. Adjoint of a linear map definition
  12. Orthonormal family definition
  13. Eigenvalue definition
  14. Spanning set definition
  15. Self-adjoint operator definition
  16. Eigenvector definition
  17. Basis definition
  18. Orthonormal basis definition
  19. Spectral theorem statement

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Spectral_theorem, Hermitian maps and Hermitian matrices
About this page
Tag06J4PXRG22
URIm2://06J4PXRG22
Kindstatement, theorem
WritingW1 stated
MSC47B15, 15A18
StatusThis copy of the site has no build data. Run m2 status 06J4PXRG22.
UsesSelf-adjoint operator, Orthonormal basis, Eigenvector
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