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Let A be a non-zero matrix that is diagonalizable. Show that A" # 0, for any positive integer r. Definition: A nilpotent matrix is a square matrix M such that M = 0 for some positive integer k. nilpotent matrices are not diagoanalizable. 0 1 Show that 0 0 1 is not diagonalizable. 0 00 Show that a non-zero n x n upper triangular matrix with zero diagonal entries is not diagonalizable.

Let A be a non-zero matrix that is diagonalizable. Show that A" # 0, for any positive integer r. Definition: A nilpotent matrix is a square matrix M such that M = 0 for some positive integer k. nilpotent matrices are not diagoanalizable. 0 1 Show that 0 0 1 is not diagonalizable. 0 00 Show that a non-zero n x n upper triangular matrix with zero diagonal entries is not diagonalizable.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let A be a non-zero matrix that is diagonalizable. Show that A" # 0, for any positive integer r. 0 for some positive integer k. Definition: A nilpotent matrix is a square matrix M such that M nilpotent matrices are not diagoanalizable.) 0 1 Show that 0 0 0 0 1 is not diagonalizable. 0 Show that a non-zero n x n upper triangular matrix with zero diagonal entries is not diagonalizable.
Transcribed Image Text:Let A be a non-zero matrix that is diagonalizable. Show that A" # 0, for any positive integer r. 0 for some positive integer k. Definition: A nilpotent matrix is a square matrix M such that M nilpotent matrices are not diagoanalizable.) 0 1 Show that 0 0 0 0 1 is not diagonalizable. 0 Show that a non-zero n x n upper triangular matrix with zero diagonal entries is not diagonalizable.
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