Math 2.0

Operator theory Proof

Proof that the eigenspaces of a self-adjoint operator are orthogonal

Builds on Inner productSelf-adjoint operatorEigenvectorEigenspaceEigenvalues of a self-adjoint operator are realOrthogonal vectors

Proof

We compute ⟨T(v),w⟩\ip{T(v)}{w} in two ways.

  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.
  2. Let vv and ww be eigenvectors with T(v)=λvT(v) = \lambda v and T(w)=μwT(w) = \mu w, where λ≠μ\lambda \ne \mu.
  3. Linearity in the first argument gives ⟨T(v),w⟩=λ⟨v,w⟩\ip{T(v)}{w} = \lambda \ip{v}{w}.
  4. Conjugate symmetry and linearity in the first argument give ⟨v,T(w)⟩=⟨v,μw⟩=μ‾⟨v,w⟩\ip{v}{T(w)} = \ip{v}{\mu w} = \overline{\mu} \ip{v}{w}.
  5. The eigenvalue μ\mu is real, so μ‾=μ\overline{\mu} = \mu.
  6. By self-adjointness, ⟨T(v),w⟩=⟨v,T(w)⟩\ip{T(v)}{w} = \ip{v}{T(w)}. Steps 3 to 5 give λ⟨v,w⟩=μ⟨v,w⟩\lambda \ip{v}{w} = \mu \ip{v}{w}.
  7. So (λ−μ)⟨v,w⟩=0(\lambda - \mu) \ip{v}{w} = 0. Since λ−μ≠0\lambda - \mu \ne 0, we get ⟨v,w⟩=0\ip{v}{w} = 0. So vv and ww are orthogonal.
  8. Now let EλE_\lambda and EμE_\mu be the eigenspaces for λ≠μ\lambda \ne \mu. Let v∈Eλv \in E_\lambda and w∈Eμw \in E_\mu.
  9. If v=0v = 0, then ⟨v,w⟩=⟨0⋅v,w⟩=0⋅⟨v,w⟩=0\ip{v}{w} = \ip{0 \cdot v}{w} = 0 \cdot \ip{v}{w} = 0. If w=0w = 0, then ⟨v,w⟩=⟨w,v⟩‾=0\ip{v}{w} = \overline{\ip{w}{v}} = 0 in the same way.
  10. Otherwise vv and ww are eigenvectors for λ\lambda and μ\mu, and step 7 gives ⟨v,w⟩=0\ip{v}{w} = 0. So EλE_\lambda and EμE_\mu are orthogonal to each other.

Idea

Moving TT across the inner product gives the factor λ\lambda on one side and μ‾=μ\overline{\mu} = \mu on the other. Two different numbers times the same inner product can agree only when the inner product is zero.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first. The path takes the proof Proof that the eigenvalues of a self-adjoint operator are real.

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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 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 operatordefinitionProof that the eigenvalues of a self-adjoint operator are real, proof, no evidence levelProof that the eigenv…proofEigenspace, definition, no evidence levelEigenspacedefinitionEigenvalues of a self-adjoint operator are real, statement, no evidence levelEigenvalues of a self…statementProof that the eigenspaces of a self-adjoint operator are orthogonal, proof, no evidence levelProof that the eigens…proof
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. 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. Orthogonal vectors definition
  9. Endomorphism definition
  10. Kernel of a linear map definition
  11. Adjoint of a linear map definition
  12. Eigenvalue definition
  13. Eigenvector definition
  14. Self-adjoint operator definition
  15. Proof that the eigenvalues of a self-adjoint operator are real proof
  16. Eigenspace definition
  17. Eigenvalues of a self-adjoint operator are real statement
  18. Proof that the eigenspaces of a self-adjoint operator are orthogonal proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://linear.axler.net/LADR4e.pdf, Theorem 7.22
About this page
TagW393MWCNHT
URIm2://W393MWCNHT
Kindproof
WritingW0 stub
MSC47B15, 15A18
ProvesEigenspaces of a self-adjoint operator are orthogonal
UsesInner product, Self-adjoint operator, Eigenvector, Eigenspace, Eigenvalues of a self-adjoint operator are real, Orthogonal vectors
Changed4 November 2026

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