Please help me understand how to do this linear algebra proof

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Discussion Overview

The discussion revolves around a linear algebra proof concerning eigenvalues and eigenvectors of a matrix A and its square A^2. Participants are exploring the implications of an eigenvalue \lambda of matrix A and how it relates to the eigenvalue of A^2 and the corresponding eigenvector.

Discussion Character

  • Homework-related

Main Points Raised

  • One participant states that if \lambda is an eigenvalue of A, then \lambda^2 is an eigenvalue of A^2 and that the eigenvector v associated with \lambda is also an eigenvector for A^2.
  • Another participant asks for clarification on the definition of an eigenvalue, suggesting that understanding this concept is key to progressing in the proof.
  • A participant acknowledges the relationship Av = \lambda v but expresses uncertainty about the next steps in the proof.
  • One participant notes that the original post has been moved to the appropriate Homework forum.

Areas of Agreement / Disagreement

Participants appear to agree on the initial definitions and relationships involving eigenvalues and eigenvectors, but there is uncertainty regarding the proof's progression and the next steps to take.

Contextual Notes

Some assumptions about the properties of eigenvalues and eigenvectors may be implicit, and the discussion does not resolve how to formally prove the statements made.

Who May Find This Useful

Students studying linear algebra, particularly those interested in eigenvalues and eigenvectors, may find this discussion relevant.

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Let A be an n x n matrix with eigenvalue [tex]\lambda[/tex] . Prove that [tex]\lambda[/tex] ^2 is and eigenvalue of A^2 and that if v is an eigenvector for A, then v is also an eigenvector for A^2.
 
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Do you understand what it means for [itex]\lambda[/itex] to be an eigenvalue of A? Just write down what it means, and you're almost done.
 
I know that Av= [tex]\lambda[/tex]v but I do not know where to go from that.
 
I see that you have reposted in the appropriate Homework forum.

Thanks!
 

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