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

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The discussion focuses on proving that if v is an eigenvector of matrices A and B with eigenvalues lambda and mui, then v is also an eigenvector of A+B and AB. The user suggests adding the equations Av = lambda*v and Bv = mui*v to derive (A+B)v = (lambda + mui)v, confirming that v is an eigenvector of A+B with eigenvalue lambda + mui. For the product AB, the user notes that multiplying AB by v allows the eigenvalues to be treated as scalars, leading to the conclusion that v is an eigenvector of AB as well. The conversation emphasizes the importance of linearity and associativity in these operations. Overall, the thread provides a clear approach to solving eigenvector problems involving matrix sums and products.
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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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