Let T : V → V be a linear map. A vector v ∈ V is...

Let T : V → V be a linear map. A vector v ∈ V is called a fixed point of T if Tv = v. For example, 0 is a fixed point for every linear map T. Show that 1 is an eigenvalue of T if and only if T has nonzero fixed points, and that these nonzero fixed points are the eigenvectors of T corresponding to eigenvalue 1

Expert Solution

If matrix representation of T is M then Tv=Mv again Tv=v implies Mv=v and since eigen value satisfies it's characteristic equation,we have that eigen value of M is 1 and since M is the matrix representation of T , therefore 1 is an eigen value of T.

Colby Messinger answered 6 months ago

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