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[M] Examine powers of a regular stochastic matrix. Display
Chapter 4, Problem 21E(choose chapter or problem)
[M] Examine powers of a regular stochastic matrix.
a. Compute \(P^k\) for k = 2, 3, 4, 5, when
\(P=\left[\begin{array}{llll}.3355 & .3682 & .3067 & .0389 \\ .2663 & .2723 & .3277 & .5451 \\ .1935 & .1502 & .1589 & .2395 \\ .2047 & .2093 & .2067 & .1765\end{array}\right]\)
Display calculations to four decimal places. What happens to the columns of \(P^k\) as k increases? Compute the steady-state vector for P.
b. Compute \(Q^k\) for k D 10, 20, . . . , 80, when
\(Q=\left[\begin{array}{rrr}.97 & .05 & .10 \\ 0 & .90 & .05 \\ .03 & .05 & .85\end{array}\right]\)
(Stability for \(Q^k\) to four decimal places may require k = 116 or more.) Compute the steady-state vector for Q. Conjecture what might be true for any regular stochastic matrix.
c. Use Theorem 18 to explain what you found in parts (a) and (b).
Questions & Answers
QUESTION:
[M] Examine powers of a regular stochastic matrix.
a. Compute \(P^k\) for k = 2, 3, 4, 5, when
\(P=\left[\begin{array}{llll}.3355 & .3682 & .3067 & .0389 \\ .2663 & .2723 & .3277 & .5451 \\ .1935 & .1502 & .1589 & .2395 \\ .2047 & .2093 & .2067 & .1765\end{array}\right]\)
Display calculations to four decimal places. What happens to the columns of \(P^k\) as k increases? Compute the steady-state vector for P.
b. Compute \(Q^k\) for k D 10, 20, . . . , 80, when
\(Q=\left[\begin{array}{rrr}.97 & .05 & .10 \\ 0 & .90 & .05 \\ .03 & .05 & .85\end{array}\right]\)
(Stability for \(Q^k\) to four decimal places may require k = 116 or more.) Compute the steady-state vector for Q. Conjecture what might be true for any regular stochastic matrix.
c. Use Theorem 18 to explain what you found in parts (a) and (b).
ANSWER:Solution 21E
(a)
the columns of the matrix converge to as
increases.
As the columns of the matrices and