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a) If v is an eigenvector of a linear operator T, corresponding to an eigenvalue A, show that v is also an eigenvector for the linear operator T2, corresponding to eigenvalue 1². (b) Even if a linear operator T has no eigenvectors, the operator T2 may have eigenvec- tors (an example is a rotation through 90° in the plane). Show that if an operator T2 has an eigenvector with a nonnegative eigenvalue λ = μ², then the operator T

a) If v is an eigenvector of a linear operator T, corresponding to an eigenvalue A, show that v is also an eigenvector for the linear operator T2, corresponding to eigenvalue 1². (b) Even if a linear operator T has no eigenvectors, the operator T2 may have eigenvec- tors (an example is a rotation through 90° in the plane). Show that if an operator T2 has an eigenvector with a nonnegative eigenvalue λ = μ², then the operator T

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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**8.** (a) \( T(T(v)) = T(\lambda v) = \lambda (\lambda v) = \lambda^2 v \). (b) \( (T^2 - \mu^2) = (T - \mu)(T + \mu) \). If \( v \neq 0 \) is such that \( (T^2 - \mu^2)(v) = 0 \) then either \( (T + \mu)(v) = 0 \) or \( (T - \mu)(w) = 0 \), where \( w = (T + \mu)(v) \) is a nonzero vector. In either case, the operator \( T \) must have an eigenvector.
Transcribed Image Text:**8.** (a) \( T(T(v)) = T(\lambda v) = \lambda (\lambda v) = \lambda^2 v \). (b) \( (T^2 - \mu^2) = (T - \mu)(T + \mu) \). If \( v \neq 0 \) is such that \( (T^2 - \mu^2)(v) = 0 \) then either \( (T + \mu)(v) = 0 \) or \( (T - \mu)(w) = 0 \), where \( w = (T + \mu)(v) \) is a nonzero vector. In either case, the operator \( T \) must have an eigenvector.
**Problem 8.** **(a)** If \( v \) is an eigenvector of a linear operator \( T \), corresponding to an eigenvalue \( \lambda \), show that \( v \) is also an eigenvector for the linear operator \( T^2 \), corresponding to eigenvalue \( \lambda^2 \). **(b)** Even if a linear operator \( T \) has no eigenvectors, the operator \( T^2 \) may have eigenvectors (an example is a rotation through 90° in the plane). Show that if an operator \( T^2 \) has an eigenvector with a nonnegative eigenvalue \( \lambda = \mu^2 \), then the operator \( T \) also has an eigenvector.
Transcribed Image Text:**Problem 8.** **(a)** If \( v \) is an eigenvector of a linear operator \( T \), corresponding to an eigenvalue \( \lambda \), show that \( v \) is also an eigenvector for the linear operator \( T^2 \), corresponding to eigenvalue \( \lambda^2 \). **(b)** Even if a linear operator \( T \) has no eigenvectors, the operator \( T^2 \) may have eigenvectors (an example is a rotation through 90° in the plane). Show that if an operator \( T^2 \) has an eigenvector with a nonnegative eigenvalue \( \lambda = \mu^2 \), then the operator \( T \) also has an eigenvector.
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