An inverse spectrum problem in linear algebra,

Question

I am reading the book, Nonnegative Matrices, by Henryk Minc, and came across an exercise that I would like to solve:

Let $$\bar \sigma = (\bar\lambda_1, ... , \bar \lambda_n)=(\lambda_1, ... \lambda_n) = \sigma $$

and

$$\max_i |\lambda_i| \in \sigma$$

Now, consider only the case $n=2$.

I want to show that there exists an entry-wise non-negative matrix of the form

$$ \begin{bmatrix} a & b \\ b & a \\ \end{bmatrix} $$

with $\sigma = (\lambda_1, \lambda_2)$ as its spectrum.

Any help is appreciated.

Thanks,

Edit: I have also read the paper "The nonnegative inverse eigenvalue problem" by Patricia Egleston, Terry Lanker and Sivaram Narayan. I am trying to solve the above problem, for the case n=2, which is supposed to be a nice, easy, warm-up to understanding the paper and the mathematical problem a bit better.

And, if this is a question that is more suitable for mathoverflow.net, please let me know.

Answer 1

With the help and guidance of @user1551, the desired matrix is

$$ \large \begin{bmatrix} \frac {\lambda_1+\lambda_2}{2} & \frac {\lambda_1-\lambda_2}{2}\\ \frac {\lambda_1-\lambda_2}{2} & \frac {\lambda_1+\lambda_2}{2} \\ \end{bmatrix} $$

Please see the comments below, if you wish to compute this matrix, too.

Thanks,