Eigenvectors and Manipulations on the Matrix

  • #1

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
139
12
If Av = xv, what is (A - I)v and (A2 + A)v?
 
  • #3
1,425
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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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