All eigenvalues of a hermitian matrix are real

Dependencies:

  1. Matrix
  2. Eigenvalues and Eigenvectors
  3. Transpose of product
  4. Matrix multiplication is associative

Let $A$ be an $n$ by $n$ hermitian matrix. Then all its eigenvalues are real.

Proof

Let $(\lambda, v)$ be an eigenvalue-eigenvector pair of $A$. \[ v^*v = \sum_{i=1}^n |v_i|^2 \ge 0 \] Since $v$ is an eigenvector, $v_i$ cannot be 0 for all $i$. Therefore, $v^*v > 0$.

\[ v^*Av = v^*(\lambda v) = \lambda v^*v \] \[ v^*Av = v^*A^*v = (Av)^*v = (\lambda v)^*v = \overline{\lambda}v^*v \] \[ \lambda v^*v = \overline{\lambda}v^*v \implies (\lambda - \overline{\lambda})v^*v = 0 \implies (\lambda - \overline{\lambda}) = 0 \implies \lambda = \overline{\lambda} \]

Since $\lambda = \overline{\lambda}$, $\lambda$ is real.

Dependency for:

  1. Bounding matrix quadratic form using eigenvalues
  2. Positive definite iff eigenvalues are positive
  3. Bound on eigenvalues of sum of matrices

Info:

Transitive dependencies:

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