hi, i have a little problem...
A. i need to prove that if v is an eigenvector which is in common for both matrices A and B, then v is also an eigenvector of AB and BA.
B. i also need to prove that if A and B have an n number of eigenverctors in common which are linear independent from each other, then AB = BA.
thanks in advance...
Let X1,X2...Xn be the n common eigenvectors, P=(X1 X2 .. Xn). P is reversable since X1..Xn are linear independent. Then AP=A(X1 ... Xn)=(AX1...AXn)=(r1X1 r2X2...rnXn)=PD1,where D1 is a diagonal matrix,r1..rn is eigenvalue. So A=PD1P^(-1).Similarily, B=PD2P^(-1).So AB=PD1D2P^(-1)=PD2D1P^(-1)=BA,since D1D2=D2D1 is very obvious.