1. Dec 11, 2012 #1

   KristenSmith

   

   1. The problem statement, all variables and given/known data
   
   Show that if A is an n x n matrix whose kth row is the same as the kth row of I_n, then 1 is an eigenvalue of A.
   
   
   2. Relevant equations None that I know of.
   
   
   
   3. The attempt at a solution I tried creating an arbitrary matrix A and set it up to to find the det(A-λI) but that got me no where.

    

   KristenSmith, Dec 11, 2012
2. 
   Phys.org - latest science and technology news stories on Phys.org

   • • Game over? Computer beats human champ in ancient Chinese game
   • • Simplifying solar cells with a new mix of materials
   • • Imaged 'jets' reveal cerium's post-shock inner strength
3. Dec 11, 2012 #2

   mfb

   

   Insights Author

   2016 Award

   Staff: Mentor

   Re: Eigenvalues
   
   If you know that A and A^t have the same eigenvalues, you can work with the original definition of eigenvalue: There is a vector x such that A^tx = 1x.

    

   mfb, Dec 11, 2012
4. Dec 11, 2012 #3

   KristenSmith

   

   Re: Eigenvalues
   
   I'm still not understanding what to do with that...

    

   KristenSmith, Dec 11, 2012
5. Dec 11, 2012 #4

   Dick

   

   Science Advisor

   Homework Helper

   Re: Eigenvalues
   
   

   Think about an expansion by minors along the kth row of A-λI. You want to show 1-λ is a factor of the characteristic polynomial.

    

   Dick, Dec 11, 2012
6. Dec 11, 2012 #5

   mfb

   

   Insights Author

   2016 Award

   Staff: Mentor

   Re: Eigenvalues
   
   

   It is easy to find a specific vector x with A^tx = 1x. That shows that x is an eigenvector of A^t with eigenvalue 1 => A^t has 1 as eigenvalue => A has 1 as eigenvalue.

    

   mfb, Dec 11, 2012

• Log in with Facebook
• Log in with Twitter

Your name or email address:

Do you already have an account?

• No, create an account now.
• Yes, my password is:
• Forgot your password?

Stay logged in