Real matrix with real eigenvalues has real eigenvectors

Dependencies:

  1. Eigenvalues and Eigenvectors
  2. Conjugation of matrices is homomorphic
  3. /complex-numbers/complex-numbers

Let $A$ be a real matrix. Let $A$ have a real eigenvalue $\lambda$ and a corresponding complex eigenvector $u$. Then a real eigenvector of $A$ exists corresponding to the eigenvalue $\lambda$.

Proof

\begin{align} & Au = \lambda u \implies \overline{A}\,\ubar = \overline{\lambda}\,\ubar \implies A\ubar = \lambda\ubar \\ &\implies A(u + \ubar) = \lambda(u + \ubar) \textrm{ and } A(i(u - \ubar)) = \lambda(i(u - \ubar)) \end{align}

$u + \ubar = 2\operatorname{Re}(u)$ and $i(u - \ubar) = -2\operatorname{Im}(u)$ are real. At least one of them must be non-zero, since $u + \ubar = u - \ubar = 0 \implies u = 0$, which contradicts the fact that $u$ is an eigenvector of $A$. Hence, one of $u + \ubar$ and $i(u - \ubar)$ is a real eigenvector of $A$ corresponding to $\lambda$.

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. /complex-numbers/complex-numbers
  5. /sets-and-relations/composition-of-bijections-is-a-bijection
  6. /sets-and-relations/equivalence-relation
  7. Group
  8. Ring
  9. Polynomial
  10. Integral Domain
  11. Comparing coefficients of a polynomial with disjoint variables
  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. Semiring
  20. Matrix
  21. Stacking
  22. System of linear equations
  23. Product of stacked matrices
  24. Matrix multiplication is associative
  25. Reduced Row Echelon Form (RREF)
  26. Conjugation of matrices is homomorphic
  27. Matrices over a field form a vector space
  28. Row space
  29. Elementary row operation
  30. Every elementary row operation has a unique inverse
  31. Row equivalence of matrices
  32. Row equivalent matrices have the same row space
  33. RREF is unique
  34. Identity matrix
  35. Inverse of a matrix
  36. Inverse of product
  37. Elementary row operation is matrix pre-multiplication
  38. Row equivalence matrix
  39. Equations with row equivalent matrices have the same solution set
  40. Basis of a vector space
  41. Linearly independent set is not bigger than a span
  42. Homogeneous linear equations with more variables than equations
  43. Rank of a homogenous system of linear equations
  44. Rank of a matrix
  45. Basis of F^n
  46. Matrix of linear transformation
  47. Coordinatization over a basis
  48. Basis changer
  49. Basis change is an isomorphic linear transformation
  50. Vector spaces are isomorphic iff their dimensions are same
  51. Canonical decomposition of a linear transformation
  52. Eigenvalues and Eigenvectors