All eigenvalues of a symmetric operator are real

Dependencies:

  1. Symmetric operator
  2. Eigenvalues and Eigenvectors
  3. Inner product is anti-linear in second argument
  4. A field is an integral domain

Let $L: V \mapsto V$ be a symmetric operator, where $V$ is a vector space over a subfield of $\mathbb{C}$.

All eigenvalues of $L$ are real.

Proof

Let $(\lambda, u)$ be an eigenvalue-eigenvector pair of $L$.

\begin{align} 0 &= \langle L(u), u \rangle - \langle u, L(u) \rangle \tag{$L$ is symmetric} \\ &= \langle \lambda u, u \rangle - \langle u, \lambda u \rangle \\ &= \lambda \langle u, u \rangle - \overline{\lambda} \langle u, u \rangle \tag{(anti-)linearity} \\ &= (\lambda - \overline{\lambda}) \langle u, u \rangle \end{align}

Since $u$ is an eigenvector, $u \neq 0$. By definiteness of inner-product, $\langle u, u \rangle \neq 0$. Since $\mathbb{C}$ is a field, it has no zero-divisors. Therefore, \[ (\lambda - \overline{\lambda}) \langle u, u \rangle = 0 \implies \lambda = \overline{\lambda} \implies \lambda \in \mathbb{R} \]

Dependency for:

  1. Orthogonally diagonalizable iff hermitian
  2. Symmetric operator on V has a basis of orthonormal eigenvectors

Info:

Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. /linear-algebra/matrices/gauss-jordan-algo
  3. /complex-numbers/conjugation-is-homomorphic
  4. /sets-and-relations/composition-of-bijections-is-a-bijection
  5. /sets-and-relations/equivalence-relation
  6. Group
  7. Ring
  8. Polynomial
  9. Integral Domain
  10. Comparing coefficients of a polynomial with disjoint variables
  11. 0x = 0 = x0
  12. Field
  13. Vector Space
  14. Linear independence
  15. Span
  16. Linear transformation
  17. Composition of linear transformations
  18. Vector space isomorphism is an equivalence relation
  19. Inner product space
  20. Inner product is anti-linear in second argument
  21. Symmetric operator
  22. A field is an integral domain
  23. Semiring
  24. Matrix
  25. Stacking
  26. System of linear equations
  27. Product of stacked matrices
  28. Matrix multiplication is associative
  29. Reduced Row Echelon Form (RREF)
  30. Matrices over a field form a vector space
  31. Row space
  32. Elementary row operation
  33. Every elementary row operation has a unique inverse
  34. Row equivalence of matrices
  35. Row equivalent matrices have the same row space
  36. RREF is unique
  37. Identity matrix
  38. Inverse of a matrix
  39. Inverse of product
  40. Elementary row operation is matrix pre-multiplication
  41. Row equivalence matrix
  42. Equations with row equivalent matrices have the same solution set
  43. Basis of a vector space
  44. Linearly independent set is not bigger than a span
  45. Homogeneous linear equations with more variables than equations
  46. Rank of a homogenous system of linear equations
  47. Rank of a matrix
  48. Basis of F^n
  49. Matrix of linear transformation
  50. Coordinatization over a basis
  51. Basis changer
  52. Basis change is an isomorphic linear transformation
  53. Vector spaces are isomorphic iff their dimensions are same
  54. Canonical decomposition of a linear transformation
  55. Eigenvalues and Eigenvectors