Real Eigen Values: Proving/Disproving Matrix AB

[asisbanerjee]

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).

 

[chiro]

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

 

[morphism]

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=...