Eigenvectors and Manipulations on the Matrix

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SUMMARY

If x is an eigenvector of matrix A, it is not necessarily an eigenvector of A -1 or A + A^2. The relationship Av = xv indicates that (A - I)v results in a transformation that does not guarantee eigenvector status. The set {v, A*v, A^2*v, A^3*v, ...} forms a T-cyclic subspace, which is a subspace of the eigenspace corresponding to the eigenvalue associated with v. Any linear combination of elements within this T-cyclic remains within the eigenspace.

PREREQUISITES
  • Understanding of eigenvectors and eigenvalues
  • Familiarity with matrix operations, specifically A - I and A + A^2
  • Knowledge of T-cyclic subspaces in linear algebra
  • Basic concepts of eigenspaces and linear combinations
NEXT STEPS
  • Study the properties of eigenvectors in relation to matrix inverses
  • Explore the implications of T-cyclic subspaces in linear transformations
  • Learn about eigenspaces and their significance in linear algebra
  • Investigate the effects of matrix addition and multiplication on eigenvectors
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Students and professionals in mathematics, particularly those focusing on linear algebra, matrix theory, and eigenvalue problems.

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If x is an eigenvector of matrix A, is it true that it is also an eigenvector of A -1, or A + A^2?

Thanks for the help.
 
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If Av = xv, what is (A - I)v and (A2 + A)v?
 
A constant multiple of an eigenvector is always an eigenvector itself. As a matter of fact, the set {v, A*v, A^2*v, A^3*v, A^4*v, A^5*v, A^6*v, A^7*v, ... } is called a T-cyclic subspace, and if v is an eigenvector of A, then the T-cyclic is a subspace of the eigenspace corresponding to the eigenvalue which v corresponds to. In particular, any linear combination of elements in the T-cyclic is inside the eigenspace.
 

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