Proving Non-Zero Eigenvalues for Rotations in Euclidean Three Space

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SUMMARY

The discussion centers on proving the existence of non-zero eigenvalues for rotations in Euclidean three-space (R³). Specifically, it addresses the scenario where a rotation operator L satisfies L(𝑣) = λ𝑣, with both λ and vector 𝑣 being non-zero. The key insight is that a full rotation of 2π radians around an axis results in eigenvalues that are not equal to zero, emphasizing the importance of the axis of rotation in this proof.

PREREQUISITES
  • Understanding of linear transformations in R³
  • Familiarity with eigenvalues and eigenvectors
  • Knowledge of rotation matrices and their properties
  • Basic concepts of vector spaces and linear algebra
NEXT STEPS
  • Study the properties of rotation matrices in R³
  • Learn how to derive eigenvalues from rotation matrices
  • Explore the geometric interpretation of eigenvectors in rotations
  • Investigate the implications of the axis of rotation on eigenvalues
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Students and educators in linear algebra, mathematicians focusing on geometric transformations, and anyone interested in the mathematical foundations of rotations in three-dimensional space.

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Help! (Euclidean three space)

Homework Statement



Given the Euclidean three space R3and if L is a rotation about the origin, can you prove a situation when L([tex]\vec{v}[/tex])=[tex]\lambda[/tex] [tex]\vec{v}[/tex] and neither lambda or vector v equal zero



Homework Equations





The Attempt at a Solution


I understand that it is a full rotation of 2 pi but I do not know exactly how to prove it.
 
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Think about the axis of rotation.
 


HallsofIvy said:
Think about the axis of rotation.

I need to see how to make the proof. I understand the concept. I just do not know how to write it down in math terms.
 

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