Suppose A and B are square matrices. In thisproblem you will show that AB and BA

Question

Suppose A and B are square matrices. In thisproblem you will show that AB and BA have thesame eigenvalues. (i) Show that if = 0 is an eigenvalue for AB,then it is also an eigenvslue for BA. (Hint: Use the fact thatdet(AB-0I)=0.)

Explanation / Answer

0 is an eigenvalue for AB => (as given in the hint) det(AB)=0.A,B are both square, so det(AB)=det(A)det(B)=0 => det(A)=0 ordet(B)=0 => 0=det(B)det(A)=det(BA) => 0 is an eigenvalue forBA as well. Again the hint says all. is an eigenvalue for AB => thereexists a non-zero vector x such that AB(x)=x. We claim that B(x) is an eigenvector for BA with eigenvalue .Indeed, BA(B(x)) = B(AB(x))=B(x)=B(x) => we arethrough.

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