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11. Definition: An n x n matrix N is said be nilpotent whenever Nk smallest such k is called the index of nilpotency of N. (a) Show that if a matrix N is nilpotent, then it must be singular. (b) Let N be a nilpotent matrix. Show that the index of nilpotentcy of N is exactly the smallest integer k such that R(Nk) = R(Nk+¹) = Onxn for some positive integer k. The

11. Definition: An n x n matrix N is said be nilpotent whenever Nk smallest such k is called the index of nilpotency of N. (a) Show that if a matrix N is nilpotent, then it must be singular. (b) Let N be a nilpotent matrix. Show that the index of nilpotentcy of N is exactly the smallest integer k such that R(Nk) = R(Nk+¹) = Onxn for some positive integer k. The

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.1: Eigenvalues And Eigenvectors
Problem 80E
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11. Definition: An n × n matrix N is said to be nilpotent whenever №k smallest such k is called the index of nilpotency of N. (a) Show that if a matrix N is nilpotent, then it must be singular. (b) Let N be a nilpotent matrix. Show that the index of nilpotentcy of N is exactly the smallest integer k such that R(N) = R(Nk+¹) Onxn for some positive integer k. The
Transcribed Image Text:11. Definition: An n × n matrix N is said to be nilpotent whenever №k smallest such k is called the index of nilpotency of N. (a) Show that if a matrix N is nilpotent, then it must be singular. (b) Let N be a nilpotent matrix. Show that the index of nilpotentcy of N is exactly the smallest integer k such that R(N) = R(Nk+¹) Onxn for some positive integer k. The
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