Prove that if t is diagonalizable then t^-1 is diagonalizable
Answers
Answered by
0
Let T be an invertible linear operator on a finite dimensional vector space V.
Given for any eigenvalue αα of T, αα^(-1) is an eigenvalue of T^(-1). I first proved that the eigenspace of Tcorresponding to eigenvalue αα is the same as the eigenspace of T^(-1) corresponding to αα^(-1). Now I need to prove that if T is diagonalizable, then T^(-1) is also diagonalizable.
I feel that given that T is invertible and diagonalizable it has a basis ββ consisting of distinct eigenvectors of T. This is where I am stuck. Would I be right in saying that T^(-1)will also have the same basis ββ? And hence prove it is diagonalizable? I only know that the eigenspaces are same for both the operators.
Given for any eigenvalue αα of T, αα^(-1) is an eigenvalue of T^(-1). I first proved that the eigenspace of Tcorresponding to eigenvalue αα is the same as the eigenspace of T^(-1) corresponding to αα^(-1). Now I need to prove that if T is diagonalizable, then T^(-1) is also diagonalizable.
I feel that given that T is invertible and diagonalizable it has a basis ββ consisting of distinct eigenvectors of T. This is where I am stuck. Would I be right in saying that T^(-1)will also have the same basis ββ? And hence prove it is diagonalizable? I only know that the eigenspaces are same for both the operators.
Similar questions