Math 2.0

Operator theory Proposition

Eigenspaces of a self-adjoint operator are orthogonal

Builds on Self-adjoint operatorEigenspaceOrthogonal vectors

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. Let vv and ww be eigenvectors of TT for distinct eigenvalues λ≠μ\lambda \ne \mu. Then vv and ww are orthogonal. So the eigenspaces of TT for distinct eigenvalues are orthogonal to each other.

Remarks

The proof: λ⟨v,w⟩=⟨T(v),w⟩=⟨v,T(w)⟩=μ⟨v,w⟩\lambda \langle v, w \rangle = \langle T(v), w \rangle = \langle v, T(w) \rangle = \mu \langle v, w \rangle, since μ\mu is real. So (λ−μ)⟨v,w⟩=0(\lambda - \mu) \langle v, w \rangle = 0, and ⟨v,w⟩=0\langle v, w \rangle = 0.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionKernel of a linear map, definition, no evidence levelKernel of a linear mapdefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionEigenvalue, definition, no evidence levelEigenvaluedefinitionEigenvector, definition, no evidence levelEigenvectordefinitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionEigenspace, definition, no evidence levelEigenspacedefinitionEigenspaces of a self-adjoint operator are orthogonal, statement, no evidence levelEigenspaces of a self…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.
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  1. Vector space definition
  2. Inner product definition
  3. Linear map definition
  4. Inner product space definition
  5. Orthogonal vectors definition
  6. Endomorphism definition
  7. Kernel of a linear map definition
  8. Adjoint of a linear map definition
  9. Eigenvalue definition
  10. Eigenvector definition
  11. Self-adjoint operator definition
  12. Eigenspace definition
  13. Eigenspaces of a self-adjoint operator are orthogonal statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Self-adjoint_operator, Properties
About this page
Tag0JHGAV9H25
URIm2://0JHGAV9H25
Kindstatement, proposition
WritingW0 stub
MSC47B15, 15A18
StatusThis copy of the site has no build data. Run m2 status 0JHGAV9H25.
UsesSelf-adjoint operator, Eigenspace, Orthogonal vectors

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