We say that two nxn matrices A and B are simultaneously diagonalizable if there exists
Chapter 7, Problem 69(choose chapter or problem)
We say that two nxn matrices A and B are simultaneously diagonalizable if there exists an invertible nxn matrix S such that S~1A S and S~1BS are both diagonal, a. Axe the matricessimultaneously diagonalizable? Explain. b. Show that if A and B are simultaneously diagonalizable then AB = BA. c. Give an example of two n x n matrices such that AB = BA, but A and B are not simultaneously diagonalizable. d. Let D be a diagonal nxn matrix with n distinct entries on the diagonal. Find all n x n matrices B that commute with D. e. Show that if AB = BA and A has n distinct eigenvalues, then A and B are simultaneously diagonalizable. (Hint: Part (d) is useful.)
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