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2. Let $A = \begin{bmatrix} .2 & 0 & 0 \ 0 & .2 & 0 \ .8 & .8 & 1 \end{bmatrix}$ and let $x = \begin{bmatrix} .3 \ .6 \ .1 \end{bmatrix}$. (x is an example of a probability vector—a vector whose entries are nonnegative and add up to 1—and A is an example of a stochastic matrix—a matrix whose columns are probability vectors) (a) Diagonalize A (i.e. find a matrix P and a diagonal matrix D such that $A = PDP^{-1}$). (b) Use your answer from part (a) to find a formula for $A^k$, where k is any positive integer. (c) Compute $A^kx$. ($A^kx$ is called a state vector) (d) A steady state vector is an eigenvector corresponding to the eigenvalue 1 that is also a probability vector. Compute $q = \lim_{k \to \infty} A^kx$ and show that q a steady state vector.

          2. Let $A = \begin{bmatrix} .2 & 0 & 0 \ 0 & .2 & 0 \ .8 & .8 & 1 \end{bmatrix}$ and let $x = \begin{bmatrix} .3 \ .6 \ .1 \end{bmatrix}$. (x is an example of a probability vector—a vector whose entries are nonnegative and add up to 1—and A is an example of a stochastic matrix—a matrix whose columns are probability vectors)
(a) Diagonalize A (i.e. find a matrix P and a diagonal matrix D such that $A = PDP^{-1}$).
(b) Use your answer from part (a) to find a formula for $A^k$, where k is any positive integer.
(c) Compute $A^kx$. ($A^kx$ is called a state vector)
(d) A steady state vector is an eigenvector corresponding to the eigenvalue 1 that is also a probability vector. Compute $q = \lim_{k \to \infty} A^kx$ and show that q a steady state vector.

2. Let A =
< b m a t r i x > and let x =
< b m a t r i x >. (x is an example of a probability vector—a vector whose entries are nonnegative and add up to 1—and A is an example of a stochastic matrix—a matrix whose columns are probability vectors)
(a) Diagonalize A (i.e. find a matrix P and a diagonal matrix D such that A = PDP^-1).
(b) Use your answer from part (a) to find a formula for A^k, where k is any positive integer.
(c) Compute A^kx. (A^kx is called a state vector)
(d) A steady state vector is an eigenvector corresponding to the eigenvalue 1 that is also a probability vector. Compute q = limk →∞ A^kx and show that q a steady state vector.

Added by Scott F.

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