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2.1.11 A matrix A E Mn(C) is called nilpotent if there exists a positive integer k such that Ak = 0. %3D (a) Show that A is nilpotent if and only if all of its eigen- values equal 0. (b) Show that if A is nilpotent then Ak = 0 for some k

2.1.11 A matrix A E Mn(C) is called nilpotent if there exists a positive integer k such that Ak = 0. %3D (a) Show that A is nilpotent if and only if all of its eigen- values equal 0. (b) Show that if A is nilpotent then Ak = 0 for some k

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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2.1.11
**2.1.20 2.1.11 A matrix A E Mn(C) is called nilpotent if there exists a positive integer k such that Ak = 0. spectral out relying rem 2.1.4) (a) Show that A is nilpotent if and only if all of its eigen- values equal 0. (b) Show that if A is nilpotent then Ak= 0 for some k<n. (a) Sho (b) Sho (c) Con tion tion ** 2.1.12 Suppose that A E M,(C) has eigenvalues M,12,..., An (listed according to algebraic multiplicity). Show that A is normal if and only if 2.1.21 A* =-A). (a) Shc (b) Shc
Transcribed Image Text:**2.1.20 2.1.11 A matrix A E Mn(C) is called nilpotent if there exists a positive integer k such that Ak = 0. spectral out relying rem 2.1.4) (a) Show that A is nilpotent if and only if all of its eigen- values equal 0. (b) Show that if A is nilpotent then Ak= 0 for some k<n. (a) Sho (b) Sho (c) Con tion tion ** 2.1.12 Suppose that A E M,(C) has eigenvalues M,12,..., An (listed according to algebraic multiplicity). Show that A is normal if and only if 2.1.21 A* =-A). (a) Shc (b) Shc
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