(a) 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. (Part (a) implies that nilpotent matrices are not diagoanalizable.) [011] (b) Show that 0 0 1 is not diagonalizable. 0 0 0 (c) Show that a non-zero n x n upper triangular matrix with zero diagonal entries is not diagonalizable.
(a) 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. (Part (a) implies that nilpotent matrices are not diagoanalizable.) [011] (b) Show that 0 0 1 is not diagonalizable. 0 0 0 (c) Show that a non-zero n x n upper triangular matrix with zero diagonal entries is not diagonalizable.
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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![(a) 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.
(Part (a) implies that nilpotent matrices are not diagoanalizable.)
[011]
(b) Show that 0 0 1 is not diagonalizable.
000
(c) Show that a non-zero n x n upper triangular matrix with zero diagonal entries is not diagonalizable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fefe71c64-8346-40ed-b3ed-a6b5092127b6%2F9cd01009-5abc-40b8-8105-5ab4d0192233%2F5q6bjg_processed.png&w=3840&q=75)
Transcribed Image Text:(a) 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.
(Part (a) implies that nilpotent matrices are not diagoanalizable.)
[011]
(b) Show that 0 0 1 is not diagonalizable.
000
(c) Show that a non-zero n x n upper triangular matrix with zero diagonal entries is not diagonalizable.
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