I'm attempting a homework problem and have an idea for proving the result that relies on whether or not matrices with positive entries have only real eigenvalues. Is this true?
For $2\times 2$ matrices this is easy to show. However I'm not sure how to decide if it is true for general $n \times n$ matrices.
I know that the Perron-Frobenius Theorem says the spectral radius of a positive matrix is itself an eigenvalue. If the result I'm asking about is true, can I use this to prove it?
Best Answer
As pointed out in this answer https://mathoverflow.net/questions/99845/properties-of-eigenvalues-of-general-nonnegative-matrices
the matrix $$ \begin{pmatrix} 0&0&0&1 \\ 1&0&0&1 \\ 0&1&0&1 \\ 0&0&1&1 \end{pmatrix} $$ has some complex eigenvalues. This remains true, if we add $0.01$ to every entry, making the matrix positive with some complex eigenvalues. (Of course, calculations have to be done by some software...)