● Matrix N € Cnxn is said to be nilpotent whenever Nk = Onxn for some positive integer k.• The smallest positive integer k such that NkOnxn, denoted by index (N) = k, is referred toas the index of nilpotency.=5. Let N be a nilpotent matrix of index k, and suppose that x is a vector such that Nk−¹x ‡ 0.Prove that the setC = {x, Nx, N²x, ..., Nk-¹x}is a linearly independent set.6. Prove that an arbitrary nonzero nilpotent matrix N is not diagonalizable.

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• Matrix N € Cnxn is said to be nilpotent whenever Nk = Onxn for some positive integer k. = • The smallest positive integer k such that Nk - as the index of nilpotency. Onxn, denoted by index (N) = k, is referred to 5. Let N be a nilpotent matrix of index k, and suppose that x is a vector such that Nk-¹x / 0. Prove that the set C = {x, Nx, N²x, ..., Nk-¹x} -1 is a linearly independent set. 6. Prove that an arbitrary nonzero nilpotent matrix N is not diagonalizable.

• Matrix N € Cnxn is said to be nilpotent whenever Nk = Onxn for some positive integer k. = • The smallest positive integer k such that Nk - as the index of nilpotency. Onxn, denoted by index (N) = k, is referred to 5. Let N be a nilpotent matrix of index k, and suppose that x is a vector such that Nk-¹x / 0. Prove that the set C = {x, Nx, N²x, ..., Nk-¹x} -1 is a linearly independent set. 6. Prove that an arbitrary nonzero nilpotent matrix N is not diagonalizable.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.9: Properties Of Determinants

Problem 34E

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