Solving Eigenvector Problems: A+B and AB with Corresponding Eigenvalues

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Homework Help Overview

The discussion revolves around eigenvector problems involving matrices A and B, specifically exploring the properties of eigenvectors and eigenvalues when considering the sum A+B and the product AB.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the relationship between the eigenvectors of A and B and their implications for A+B and AB. The original poster attempts to establish a connection between the eigenvalue equations and seeks hints for progressing further.

Discussion Status

Some participants have provided guidance on how to combine the eigenvalue equations for A and B, suggesting that adding the equations could lead to insights about the eigenvalues of A+B. There is also mention of the need to consider properties like associativity and linearity. The discussion about finding the eigenvalues of AB is ongoing, with hints provided but no consensus reached.

Contextual Notes

Participants are navigating through the implications of eigenvalues and eigenvectors without providing complete solutions, reflecting the constraints of the homework context.

Jennifer1990
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Homework Statement


Suppose that v is an eigenvector of both A and B with corresponding eigenvalues lambda and mui respectively. Show that v is an eigenvector of A+B and of AB and determine the corresponding eigenvalues


Homework Equations





The Attempt at a Solution


Av = lambda*v
Bv = mui*v
this is all i can think of...can someone give me a hint abt the next step?
 
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Add the two equations together, tada! Of course, you'll need to exploit associativity... or linearity... man I am always getting terms confused.
 
Last edited:


u mean like:
(Av +Bv) = lambda*v + mui*v
(A+B)v = (lambda + mui) v
 


Yup! As for finding the eigenvalues of AB, simply multiply AB by v and remember that your eigenvalues are scalars that can move freely.
 

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