Eigenvectors, Eigenvalues and Idempotent

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SUMMARY

An n x n matrix A is defined as idempotent if A^2 = A. In the context of eigenvalues, it is established that if λ is an eigenvalue of an idempotent matrix, then λ must be either 0 or 1. This conclusion arises from the relationship A*v = λ*v, where v is a nonzero vector. By applying the matrix A again and rearranging the terms, one can derive the possible values for λ.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically eigenvalues and eigenvectors.
  • Familiarity with matrix operations, particularly idempotent matrices.
  • Knowledge of the properties of eigenvalues in relation to matrix transformations.
  • Basic skills in manipulating algebraic expressions involving matrices.
NEXT STEPS
  • Study the properties of idempotent matrices in linear algebra.
  • Learn about the implications of eigenvalues in matrix theory.
  • Explore the derivation of eigenvalues from characteristic polynomials.
  • Investigate the applications of eigenvalues and eigenvectors in various fields such as data science and machine learning.
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Students and professionals in mathematics, particularly those focusing on linear algebra, as well as data scientists and engineers working with matrix computations and transformations.

mpm
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I have a question that deals with all three of the terms in the title. I'm not really even sure where to begin on this. I was hoping someone could help.

Question:

An n x n matrix A is said to be idempotent if A^2 = A. Show that if λ is an eigenvalue of an independent matrix, then λ must either be 0 or 1.

If I could get some help on this I would really appreciate it.

Thanks,

mpm
 
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Well you know that there must exist a nonzero vector v such that A*v=lamda*v. Now play with this statement by applying A again, and rearanging terms so that you end up with only expressions involving lambda. Then solve for lambda.
 

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