Type in Latex **Problem**. Let $$A = \begin{bmatrix} .5 & .2 & .3 \\ .3 & .8 & .3 \\ .2 & 0 & .4 \end{bmatrix}.$$ This matrix is an example of a **stochastic matrix**: its column sums are all equal to 1. The vectors $$\mathbf{v}_1 = \begin{bmatrix} .3 \\ .6 \\ .1 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}$$ are all eigenvectors of $A$. * Compute $\left[\begin{array}{rrr} 1 & 1 & 1 \end{array}\right]\cdot\mathbf{x}_0$ and deduce that $c_1 = 1$. * Finally, let $\mathbf{x}_k = A^k \mathbf{x}_0$. Show that $\mathbf{x}_k \longrightarrow \mathbf{v}_1$ as $k$ goes to infinity. (The vector $\mathbf{v}_1$ is called a **steady-state vector** for $A.$) **Solution**. To prove that $c_1 = 1$, we first left-multiply both sides of the above equation by $[1 \, 1\, 1]$ and then simplify both sides: $$ \begin{aligned} [1 \, 1\, 1]\mathbf{x}_0 &= [1 \, 1\, 1](c_1\mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3\mathbf{v}_3) \\ &= [1 \, 1\, 1] \\ &= \\ \end{aligned} $$
Type in Latex
**Problem**. Let $$A = \begin{bmatrix} .5 & .2 & .3 \\ .3 & .8 & .3 \\ .2 & 0 & .4 \end{bmatrix}.$$ This matrix is an example of a **stochastic matrix**: its column sums are all equal to 1. The
* Compute $\left[\begin{array}{rrr} 1 & 1 & 1 \end{array}\right]\cdot\mathbf{x}_0$ and deduce that $c_1 = 1$.
* Finally, let $\mathbf{x}_k = A^k \mathbf{x}_0$. Show that $\mathbf{x}_k \longrightarrow \mathbf{v}_1$ as $k$ goes to infinity. (The vector $\mathbf{v}_1$ is called a **steady-state vector** for $A.$)
**Solution**.
To prove that $c_1 = 1$, we first left-multiply both sides of the above equation by $[1 \, 1\, 1]$ and then simplify both sides:
$$
\begin{aligned}
[1 \, 1\, 1]\mathbf{x}_0 &= [1 \, 1\, 1](c_1\mathbf{v}_1 + c_2 \mathbf{v}_2 + c_3\mathbf{v}_3) \\
&= [1 \, 1\, 1] \\
&= \\
\end{aligned}
$$
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