Math 2.0

Operator theory Theorem

Eigenvalues of a self-adjoint operator are real

Builds on Self-adjoint operatorEigenvalue

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 each eigenvalue of TT is a real number.

Remarks

If T(v)=λvT(v) = \lambda v with v≠0v \ne 0, then λ⟨v,v⟩=⟨T(v),v⟩=⟨v,T(v)⟩=λˉ⟨v,v⟩\lambda \langle v, v \rangle = \langle T(v), v \rangle = \langle v, T(v) \rangle = \bar\lambda \langle v, v \rangle, so λ=λˉ\lambda = \bar\lambda. In particular, a Hermitian matrix has only real eigenvalues. So a real symmetric matrix has only real eigenvalues, also when we view it as a complex matrix.

Reading path

To understand this page from the ground up, read these 9 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 spacedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionEigenvalues of a self-adjoint operator are real, statement, no evidence levelEigenvalues 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.
  • no evidence level
  1. Vector space definition
  2. Inner product definition
  3. Linear map definition
  4. Inner product space definition
  5. Endomorphism definition
  6. Adjoint of a linear map definition
  7. Eigenvalue definition
  8. Self-adjoint operator definition
  9. Eigenvalues of a self-adjoint operator are real statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Self-adjoint_operator, Properties
About this page
Tag0C4NT4GT3X
URIm2://0C4NT4GT3X
Kindstatement, theorem
WritingW0 stub
MSC47B15, 15A18
StatusThis copy of the site has no build data. Run m2 status 0C4NT4GT3X.
UsesSelf-adjoint operator, Eigenvalue

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