Problem 3. Let A, B be n × n matrices. The following are two incorrect proofs that AB has the same non-zero eigenvalues as BA. For each, state two things wrong with the proof: (i) We will prove that AB and BA have the same characteristic equation. We have that det(AB - ?I) = det(AB A A^{-1} - ? A A^{-1}) = det(A(BA - ?I)A^{-1}) = det(A) + det(BA - ?I) - det(A) = det(BA - ?I) Hence det(AB - ?I) = det(BA - ?I), and so AB and BA have the same eigenvalues. (ii) Suppose not. We will prove by contradiction. Let ?, ? be non-zero eigenvalues of AB, BA respectively such that ? ? ?. Then we have that for an eigenvector v? AB v? = ? v?, BA v? = ? v? ? (AB - BA) v? = (? - ?) v? ? 0 v? = (? - ?) v? ? ? - ? = 0 ? ? = ? Hence we have arrived at a contradiction. So AB and BA have the same eigenvalues.
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The correct calculation should be det(AB^T) = det(AB) = det(BA). ** Show more…
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