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24. Markov Matrices; Fourier Series

markov $A = \begin {bmatrix} 0.01 & 0.5 \ 0.99 &0.5 \ \end {bmatrix}$

All numbers $≥ 0$
$\forall$ columns $\sum$ of elements $= 1$
$\lambda_1 = 1$ is eigenvalue, $\forall i$ $|\lambda_i| < 1$
$x_1 \geq 0$

All columns of $A - I = 0 ⇒ A-I$ singular because rows dependent $A^T\times (1,1,1)^T = 0$

$\lambda$ of $A$ and $A^T$ same

$U_{k+1} = A\times U_k$