Suppose S T LV Prove that 0 is an eigenvalue of ST if and o

Suppose S, T L(V ). Prove that 0 is an eigenvalue of ST if and only if 0 is an eigenvalue of T S.

Solution

Suppose v not equal to 0 and ST v = v.
Multiply by T to get
T S(T v) = T v.
Thus if T v not equal = 0 then is also an eigenvalue of T S, with non-zero eigenvector
T v.
On the other hand, if T v = 0, then = 0 is an eigenvalue of ST. But if
T is not invertible, then rangeT S rangeT is not equal to V, so T S has a
non-trivial null space, hence 0 is an eigenvalue of T S