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## Homework Statement

Let U be a fixed nxn matrix and consider the operator T: Msub(n,n)------>Msub(n,n)

given by T(A)=UA.

Show that c is an eigenvalue of T if and only if it is an eigenvalue of U.

## Homework Equations

## The Attempt at a Solution

If T(A)=UA then T(A)-UA=0 (T-U)A=0.

Let v be an eigenvector of T so Tv=cv.

If v is an eigenvector[tex]\in[/tex]A then A

is not a zero matrix so for (T-U)A=0 we

have Tv-Uv=0 so cv-Uv=0

Uv=cv so c must be an eigenvalue of U for

Uv=Tv=cv and for Uv-Tv=0 for v[tex]\in[/tex]A.