4. Let A and B be n x n matrices with AB = BA, and let |v) E C" be an eigenvector of A with corresponding eigenvalue A E C. (a) Prove that A|w) = A |w), where |w) = B |v). (That is, prove that Jw) E Sa(A).) (b) If we further assume that A is Hermitian and non-degenerate, show that |v) is also an eigenvector of B. (The corresponding eigenvalue in this case is, in general, different than A.)
4. Let A and B be n x n matrices with AB = BA, and let |v) E C" be an eigenvector of A with corresponding eigenvalue A E C. (a) Prove that A|w) = A |w), where |w) = B |v). (That is, prove that Jw) E Sa(A).) (b) If we further assume that A is Hermitian and non-degenerate, show that |v) is also an eigenvector of B. (The corresponding eigenvalue in this case is, in general, different than A.)
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Transcribed Image Text:4. Let A and B be n x n matrices with AB =
BA, and let |v) E C" be an eigenvector of A with
corresponding eigenvalue A E C.
(a) Prove that A |w) = A |w), where |w) = B |v). (That is, prove that |w) E Sx(A).)
(b) If we further assume that A is Hermitian and non-degenerate, show that |v) is also an eigenvector
of B. (The corresponding eigenvalue in this case is, in general, different than A.)
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