Question

Suppoese T is self-adjoint. Show that T^2(v)=0 implies T(v)=0. Using this prove that T^n(v)=0 also implies that T(v)=0 for n>0

Answer 1

if T(v)#0 -> T*(v)=T(v)#0 ->T^2(v)=T(v).T*(v)#0

Answer 2

because T.T*=detT.I & detT #0

Answer 3

induction:
for n=2 is ok
suppose for k is true
then try to show for k+1 is true

Answer 4

why detT#0 ?

Answer 5

because detT*=detT^m-1=detT -> detT^m=1 or detT#0

Answer 6

if m=odd detT=1 and if m=even detT=+-1

Answer 7

done?