Eigenpair of power of a matrix

Dependencies:

  1. Matrix
  2. Eigenvalues and Eigenvectors
  3. Identity matrix
  4. Matrix multiplication is associative
  5. Inverse of product

If $(\lambda, v)$ is an eigenpair of matrix $A$, then $(\lambda^k, v)$ is an eigenpair of $A^k$, where $k \in \mathbb{Z}$.

Proof

Let $P(k)$ be the claim that $(\lambda^k, v)$ is an eigenpair of $A^k$. We'll prove $P(k)$ for all $k \ge 0$ by mathematical induction.

Base case: $P(1)$ is trivially true. When $k = 0$, $A^k = I$ and $Iv = v = \lambda^0 v$. So $(\lambda^k, v)$ is an eigenpair of $A^k$. Hence $P(0)$.

Inductive step: Assume $P(k)$ is true for $k \ge 1$. \begin{align} & A^{k+1}v = (AA^k)v = A(A^kv) = A(\lambda^k v) \\ &= \lambda^k (Av) = \lambda^k (\lambda v) = \lambda^{k+1}v \end{align} Hence, $P(k+1)$ is true. By mathematical induction, $P(k)$ is true $\forall k \ge 0$.

If $A$ is invertible, then \[ Av = \lambda v \implies v = A^{-1}(\lambda v) = \lambda (A^{-1}v) \implies A^{-1}v = \lambda^{-1}v \]

Since $A^{-k} = (A^k)^{-1}$, $((\lambda^{k})^{-1}, v) = (\lambda^{-k}, v)$ is an eigenpair of $A$.

Dependency for: None

Info:

Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. /linear-algebra/matrices/gauss-jordan-algo
  3. /sets-and-relations/composition-of-bijections-is-a-bijection
  4. /sets-and-relations/equivalence-relation
  5. Group
  6. Ring
  7. Polynomial
  8. Integral Domain
  9. Comparing coefficients of a polynomial with disjoint variables
  10. Field
  11. Vector Space
  12. Linear independence
  13. Span
  14. Linear transformation
  15. Composition of linear transformations
  16. Vector space isomorphism is an equivalence relation
  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. Matrices over a field form a vector space
  25. Row space
  26. Elementary row operation
  27. Every elementary row operation has a unique inverse
  28. Row equivalence of matrices
  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
  49. Eigenvalues and Eigenvectors