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$, why can $M(a,b)$ always be diagonalized? I'm confused as to why this is the case, and wouldn't this mean that higher powers of $M$ will always approach a rank one matrix?
2026-03-26 06:20:13.1774506013
Diagonalization of Markov matrices
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