Real eigen values

  • #1
I need to either prove or disprove by a counterexample the following proposition:
" Let A be an m by n row-stochastic matrix in which all entries are positive real numbers and let B be an n by m column-stochastic matrix with the same feature. Then all eigen values of the m by m matrix AB are real."
Can anyone help? It may be noted that AB is not necessarily symmetric (Hermitian).
 
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Answers and Replies

  • #2
chiro
Science Advisor
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I need to either prove or disprove by a counterexample the following proposition:
" Let A be an m by n row-stochastic matrix in which all entries are positive and let B be an n by m column-stochastic matrix with the same feature. Then all eigen values of the m by m matrix AB are real."
Can anyone help?
Hello asisbanerjee and welcome to the forums.

Maybe this will give you a headstart:

http://en.wikipedia.org/wiki/Positive-definite_matrix
 
  • #3
morphism
Science Advisor
Homework Helper
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I need to either prove or disprove by a counterexample the following proposition:
" Let A be an m by n row-stochastic matrix in which all entries are positive real numbers and let B be an n by m column-stochastic matrix with the same feature. Then all eigen values of the m by m matrix AB are real."
Can anyone help? It may be noted that AB is not necessarily symmetric (Hermitian).
Do you know an example of a (square) stochastic matrix A whose entries are positive but which has a nonreal eigenvalue?

That's enough to find a counterexample, because you can then take B=...
 
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