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= 5. A linear operator T: V → V is said to be nilpotent if Tk 0 for some positive integer k. Similarly, a square matrix A is said to be nilpotent if Ak = 0 for some positive integer k. (a) Let T: V→ V be a linear operator on a finite-dimensional complex inner product space V. Show that if T is normal and nilpotent, then T = 0. (Hint: spectral theorem for normal operators.) (b) Use (a) to show that if a square matrix A is normal and nilpotent, then A = 0. (Hint: apply (a) to T = LA: Cn Cn.)

= 5. A linear operator T: V → V is said to be nilpotent if Tk 0 for some positive integer k. Similarly, a square matrix A is said to be nilpotent if Ak = 0 for some positive integer k. (a) Let T: V→ V be a linear operator on a finite-dimensional complex inner product space V. Show that if T is normal and nilpotent, then T = 0. (Hint: spectral theorem for normal operators.) (b) Use (a) to show that if a square matrix A is normal and nilpotent, then A = 0. (Hint: apply (a) to T = LA: Cn Cn.)

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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- 5. A linear operator T: V → V is said to be nilpotent if Th 0 for some positive integer k. Similarly, a square matrix A is said to be nilpotent if Ak = 0 for some positive integer k. = (a) Let T: V → V be a linear operator on a finite-dimensional complex inner product space V. Show that if T is normal and nilpotent, then T = 0. (Hint: spectral theorem for normal operators.) (b) Use (a) to show that if a square matrix A is normal and nilpotent, then A = 0. (Hint: apply (a) to T = LA: Cn Cn.) →
Transcribed Image Text:- 5. A linear operator T: V → V is said to be nilpotent if Th 0 for some positive integer k. Similarly, a square matrix A is said to be nilpotent if Ak = 0 for some positive integer k. = (a) Let T: V → V be a linear operator on a finite-dimensional complex inner product space V. Show that if T is normal and nilpotent, then T = 0. (Hint: spectral theorem for normal operators.) (b) Use (a) to show that if a square matrix A is normal and nilpotent, then A = 0. (Hint: apply (a) to T = LA: Cn Cn.) →
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