An n x n matrix A is called nilpotent if there is a positive integer k such that Ak = 0. Show that a nonzero nilpotent matrix is not diagonalizable. [Hint: if A is diagonalizable then P-¹APD where P is invertible and D is a diagonal matrix. Solve this for. A and raise to power k. ] =

Elementary Linear Algebra (MindTap Course List)
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Chapter7: Eigenvalues And Eigenvectors
Section7.1: Eigenvalues And Eigenvectors
Problem 80E
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6. An n x n matrix A is called nilpotent if there is a positive integer k such that Ak = 0.
Show that a nonzero nilpotent matrix is not diagonalizable. [Hint: if A is diagonalizable
then P-¹AP = D where P is invertible and D is a diagonal matrix. Solve this for. A
and raise to power k. ]
Transcribed Image Text:6. An n x n matrix A is called nilpotent if there is a positive integer k such that Ak = 0. Show that a nonzero nilpotent matrix is not diagonalizable. [Hint: if A is diagonalizable then P-¹AP = D where P is invertible and D is a diagonal matrix. Solve this for. A and raise to power k. ]
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