i am told that v is an eigenvector of A with eigenvalue p. I am to show that p is an eigenvalue of A^3 -4A^2 + I, and find it's eigenvalue.
I have first shown that p^3 is an eigenvalue of A^3 by manipulating the equation: Av = pv (as v has eigenvalue p)
so i multiplied both sides by A^2, and ultimately reached the end result of: A^3v = p^3v, and thus p^3 is an eigenvalue of A^3.
I did the same process again but instead multiplied Av = pv by A to show that p^2 is an eigenvalue of A^2.
I was wondering, will this show that v is an eigenvector of A^3 -4A^2 + I? I have shown that p^3 is an eigenvalue of A^3 and p^2 is an eigenvalue of A^2 by playing with the Av = pv relationship.
I appreciate any help, thank you.