Eigvector/value problem (need validation of my proof)

[rock.freak667]

[SOLVED] eigvector/value problem (need validation of my proof)

Homework Statement

Show that if $\lambda$ is an e.value of a square matrix A with e as a corresponding e.vector, and [ites]\mu[/itex] is an e.value of the square matrix B for which e is also a corresponding e.vector,the $\lambda + \mu$ is an e.value of the matrix A+B with e as a corresponding e.vector

Homework Equations

The Attempt at a Solution

From the def'n of an e.vector

$$Ax= \lambda x$$

$$Ax+Bx= \lambda x + \mu x$$

$$(A+B)x= (\lambda +\mu)x$$

hence $\lambda +\mu$ is an e.value of A+B

 

[foxjwill]

I don't see anything wrong with your proof. Although since the problem defines $$e$$ as the eigenvector, it'd probably be more prudent (albeit arbitrary) to use that instead of $$x$$.

 

[Hurkyl]

I don't see anything wrong with your proof. ... it'd probably be more prudent (albeit arbitrary) to use that instead of $$x$$.

I would consider the fact he used x when he should have been using e an actual error in his proof. Yes, it might simply be a persistent typographical error (which is still an error!) -- but it might also be a symptom of a deeper misunderstanding of how to manipulate mathematical statements.

However, I do completely agreement with you that the original poster's proof idea is correct.

 

[rock.freak667]

ah yeah, a typo, supposed to be e since that is the e.vector.

 

[Hurkyl]

Ah, there we go. Mystery solved.