Math 2.0

Operator theory Proof

Proof that the eigenvalues of a self-adjoint operator are real

Builds on Inner productSelf-adjoint operatorEigenvalueComplex conjugate

Proof

We compute ⟨T(v),v⟩\ip{T(v)}{v} in two ways for an eigenvector vv.

  1. Let VV be a finite-dimensional inner product space over 𝕂=ℝ\mathbb{K} = \R or ℂ\C, with inner product ⟨⋅,⋅⟩\ip{\cdot}{\cdot}. Let T ⁣:V→VT \colon V \to V be self-adjoint, so ⟨T(v),w⟩=⟨v,T(w)⟩\ip{T(v)}{w} = \ip{v}{T(w)} for all v,w∈Vv, w \in V.
  2. If 𝕂=ℝ\mathbb{K} = \R, every scalar is real, so there is nothing to prove. Let 𝕂=ℂ\mathbb{K} = \C.
  3. Let λ\lambda be an eigenvalue of TT. So there is v∈Vv \in V with v≠0v \ne 0 and T(v)=λvT(v) = \lambda v.
  4. Linearity in the first argument gives ⟨T(v),v⟩=⟨λv,v⟩=λ⟨v,v⟩\ip{T(v)}{v} = \ip{\lambda v}{v} = \lambda \ip{v}{v}.
  5. Conjugate symmetry and linearity in the first argument give ⟨v,λv⟩=λ⟨v,v⟩‾=λ‾ ⟨v,v⟩\ip{v}{\lambda v} = \overline{\lambda \ip{v}{v}} = \overline{\lambda}\, \ip{v}{v}, because ⟨v,v⟩\ip{v}{v} is real.
  6. By step 1, ⟨T(v),v⟩=⟨v,T(v)⟩=⟨v,λv⟩\ip{T(v)}{v} = \ip{v}{T(v)} = \ip{v}{\lambda v}. So steps 4 and 5 give λ⟨v,v⟩=λ‾⟨v,v⟩\lambda \ip{v}{v} = \overline{\lambda} \ip{v}{v}.
  7. Since v≠0v \ne 0, positivity gives ⟨v,v⟩>0\ip{v}{v} > 0. Dividing by it gives λ=λ‾\lambda = \overline{\lambda}.
  8. A complex number equal to its complex conjugate has imaginary part 00. So λ\lambda is real.

Idea

Self-adjointness lets TT move from one side of the inner product to the other. On an eigenvector, moving TT to the left gives the factor λ\lambda, and moving it to the right gives λ‾\overline{\lambda}. So λ=λ‾\lambda = \overline{\lambda}.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectVector space, definition, no evidence levelVector spacedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionComplex conjugate, definition, no evidence levelComplex conjugatedefinitionInner 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 operatordefinitionProof that the eigenvalues of a self-adjoint operator are real, proof, no evidence levelProof that the eigenv…proof
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  1. Complex numbers object
  2. Vector space definition
  3. Real and imaginary parts definition
  4. Inner product definition
  5. Linear map definition
  6. Complex conjugate definition
  7. Inner product space definition
  8. Endomorphism definition
  9. Adjoint of a linear map definition
  10. Eigenvalue definition
  11. Self-adjoint operator definition
  12. Proof that the eigenvalues of a self-adjoint operator are real proof

Where this leads

36 pages build on this one.

Further reading

  1. Web https://linear.axler.net/LADR4e.pdf, Theorem 7.12
About this page
Tag7NSB724DV3
URIm2://7NSB724DV3
Kindproof
WritingW0 stub
MSC47B15, 15A18
ProvesEigenvalues of a self-adjoint operator are real
UsesInner product, Self-adjoint operator, Eigenvalue, Complex conjugate
Changed4 November 2026

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