Eigenvector / eigenvalue pairs for a Markov Matrix

Question

If we're given a $ \displaystyle 2 \times 2 $ Markov Matrix (so all entries are non-negative and columns add to 1) M$(a,b)$ such that $$M = M(a,b) := \begin{bmatrix}1 - a & b\\a & 1 - b \end{bmatrix}$$ where $a$ and $b$ are $ 0 ≤ a ≤ 1, 0 ≤ b ≤ 1$, I know that $λ1 = 1$ is an eigenvalue for $M(a,b)$, but how would I find a corresponding eigenvector $u_{1}(a, b)$ such that when normalized, $e^{T}u1(a, b) = 1$? There should also be a second eigenvalue / eigenvector right?

Answer 1

The standard linear algebra techniques will work. The eigenvalues are the roots of the characteristic polynomial. An eigenvector for eigenvalue $\lambda$ is in the null space of $M - \lambda I$. You can use Gaussian elimination to find one, then multiply by a suitable constant to get the desired normalization.