Eigenvalues and Eigenvectors

Dependencies:

  1. Canonical decomposition of a linear transformation
  2. Matrix of linear transformation

Let $V$ be a vector space over a field F. Let $T: V \mapsto V$ be a linear transformation.

If $T(x) = \lambda x$ is true for some $\lambda \in F$ and some $x \neq 0 \in V$, then $\lambda$ is called an eigenvalue of $T$ and $x$ is called an eigenvector of $T$. $(\lambda, x)$ is called an eigenpair of $T$.

If $V$ is finite-dimensional and $\dim(V) = n$, the eigenvalues and eigenvectors of $T$ can be found by finding the eigenvalues and eigenvectors of another transformation $T_2: F^n \mapsto F^n$ where $T_2(x) = Ax$ and $A$ is a square matrix over $F$.

Consequently, the eigenvalues and eigenvectors of a matrix $A$ are defined as solutions to $Ax = \lambda x$.

Note that for a given eigenvector, there can be exactly one eigenvalue.

Proof

Let $\dim(V) = n$. Let $T = T_1^{-1}T_2T_1$ be the canonical decomposition of $T$, where $T_1: V \mapsto F^n$ and $T_1$ is an isomorphism.

\[ (T_1^{-1}T_2T_1)(v) = \lambda v \iff T_2(T_1(v)) = T_1(\lambda v) \iff T_2(T_1(v)) = \lambda T_1(v) \]

Therefore, $(\lambda, v)$ is a solution to $T(v) = \lambda v$ iff $(\lambda, T_1(v))$ is a solution to $T_2(x) = \lambda x$. Therefore, $T$ and $T_2$ have the same eigenvalues. Eigenvectors of $T$ can be found by applying $T_1^{-1}$ on eigenvectors of $T_2$.

Every linear transformation from $F^n$ to $F^n$ can be expressed as matrix pre-multiplication. So $\exists A, T_2(x) = Ax$. Therefore, the eigenvalue equation becomes $Ax = \lambda x$.

Suppose the eigenvector $v$ has at least 2 eigenvalues $\lambda_1$ and $\lambda_2$ where $\lambda_1 \neq \lambda_2$. \[ T(v) = \lambda_1 v = \lambda_2 v \implies (\lambda_1 - \lambda_2) v = 0 \implies v = 0 \implies \bot \]

Dependency for:

  1. Bounding matrix quadratic form using eigenvalues
  2. Positive definite iff eigenvalues are positive
  3. Characteristic polynomial of a matrix
  4. Real matrix with real eigenvalues has real eigenvectors
  5. Eigenspace
  6. Bound on eigenvalues of sum of matrices
  7. Symmetric operator on V has a basis of orthonormal eigenvectors
  8. Eigenpair of affine transformation
  9. Eigenvectors of distinct eigenvalues are linearly independent
  10. Every complex matrix has an eigenvalue
  11. Eigenpair of power of a matrix
  12. All eigenvalues of a symmetric operator are real
  13. All eigenvalues of a hermitian matrix are real
  14. Diagonalization

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. Reduced Row Echelon Form (RREF)
  24. Elementary row operation
  25. Every elementary row operation has a unique inverse
  26. Row equivalence of matrices
  27. Matrices over a field form a vector space
  28. Row space
  29. Row equivalent matrices have the same row space
  30. RREF is unique
  31. Identity matrix
  32. Inverse of a matrix
  33. Inverse of product
  34. Elementary row operation is matrix pre-multiplication
  35. Row equivalence matrix
  36. Equations with row equivalent matrices have the same solution set
  37. Basis of a vector space
  38. Linearly independent set is not bigger than a span
  39. Homogeneous linear equations with more variables than equations
  40. Rank of a homogenous system of linear equations
  41. Rank of a matrix
  42. Basis of F^n
  43. Matrix of linear transformation
  44. Coordinatization over a basis
  45. Basis changer
  46. Basis change is an isomorphic linear transformation
  47. Vector spaces are isomorphic iff their dimensions are same
  48. Canonical decomposition of a linear transformation