For a reversible discrete-time Markov process, if given a stationary distribution and a distinct set of eigenvalues, is the transition matrix unique?

Question

Another way of phrasing my question: if given a stationary distribution ($\vec{\pi}$) and spectrum ($\vec{\lambda}$) of a transition matrix ($P_{ij}$) for an irreducible and reversible discrete-time process, how many different transition matrices could one construct?

For the 2-dimensional case this appears to be true, since $P^n_{11}= \pi_1 + \pi_2\lambda^n_2$.

If the transition matrix is unique, is there an algorithm to generate the transition matrix from $\vec{\pi}$ and $\vec{\lambda}$?