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Eigenvectors and Manipulations on the Matrix

  1. Apr 28, 2009 #1
    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.
  2. jcsd
  3. Apr 28, 2009 #2
    If Av = xv, what is (A - I)v and (A2 + A)v?
  4. Apr 29, 2009 #3
    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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