: Linear Algebra: Let B = (S^-1)AS, and x be an eigenvector of B belonging to an eigenvalue L. Show that Sx is an eigenvector of A belonging to L.

Question

: Linear Algebra: Let B = (S^-1)AS, and  x be an eigenvector of B belonging to an eigenvalue L. Show that Sx is an eigenvector of A belonging to L.

anonymous · Answer

The setup to this question indicates that
$$Bx = Lx$$
where L is a constant and B is the matrix
$$B = (S^{-1}AS)$$
Which means that
$$A = SBS^{-1}$$
It follows that
$$A(Sx) = SBS^{-1}(Sx) = SB(x) = S(Lx) = L(Sx) $$
So Sx is indeed an eigenvector of A with eigenvalue L.

anonymous · Answer

Thank you, that reminded me to use the fact that Bx = Lx, which I forget so easily in the middle of all these hw..

anonymous · Answer

No problem :)