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2. Let A = egin{bmatrix} .2 & 0 & 0 \ 0 & .2 & 0 \ .8 & .8 & 1 end{bmatrix} and let x = egin{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^{k}x. (A^{k}x 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 o infty} A^{k}x and show that q a steady state vector. (e) Given a stochastic matrix P and probability vector x, the sequence of vectors P^{k}x is called a Markov Chain. It is a general fact that any Markov Chain P^{k}x approaches a steady state vector as k goes to infty. For more details, watch the Chapter 5 supplemental videos on Markov Chains on the class webpage. (There is nothing to do for this part of the problem)

          2. Let A = egin{bmatrix} .2 & 0 & 0 \ 0 & .2 & 0 \ .8 & .8 & 1 end{bmatrix} and let x = egin{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^{k}x. (A^{k}x 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 	o infty} A^{k}x and show that q a steady state vector.
(e) Given a stochastic matrix P and probability vector x, the sequence of vectors P^{k}x is called a Markov Chain. It is a general fact that any Markov Chain P^{k}x approaches a steady state vector as k goes to infty. For more details, watch the Chapter 5 supplemental videos on Markov Chains on the class webpage. (There is nothing to do for this part of the problem)

2. Let A = eginbmatrix .2 0 0 0 .2 0 .8 .8 1 endbmatrix and let x = eginbmatrix .3 .6 .1 endbmatrix. (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 o infty A^kx and show that q a steady state vector.
(e) Given a stochastic matrix P and probability vector x, the sequence of vectors P^kx is called a Markov Chain. It is a general fact that any Markov Chain P^kx approaches a steady state vector as k goes to infty. For more details, watch the Chapter 5 supplemental videos on Markov Chains on the class webpage. (There is nothing to do for this part of the problem)

Added by Matthew E.

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