Eigenvectors and Manipulations on the Matrix

[sleepisgood]

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.

 

[VKint]

If Av = xv, what is (A - I)v and (A² + A)v?

 

[Werg22]

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.