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1-α B Show that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written in the form P = a 1-B where and are constants between 0 and 1. (There are two linearly independent steady-state vectors if a=B=0. Otherwise, there is only one.) If P is a stochastic matrix, then a steady-state vector, or equilibrium vector, for P is a probability vector q such that Pq=q. Pq=1. Pq=0. Pq=P.

1-α B Show that every 2 x 2 stochastic matrix has at least one steady-state vector. Any such matrix can be written in the form P = a 1-B where and are constants between 0 and 1. (There are two linearly independent steady-state vectors if a=B=0. Otherwise, there is only one.) If P is a stochastic matrix, then a steady-state vector, or equilibrium vector, for P is a probability vector q such that Pq=q. Pq=1. Pq=0. Pq=P.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Show that every 2x2 stochastic matrix has at least one steady-state vector. Any such matrix can be written in the form P = 1-α B α 1-B where x and ẞ are constants between 0 and 1. (There are two linearly independent steady-state vectors if a = ß= 0. Otherwise, there is only one.) If P is a stochastic matrix, then a steady-state vector, or equilibrium vector, for P is a probability vector q such that Pq=q Pq=1. 8] Pq=0. Pq=P.
Transcribed Image Text:Show that every 2x2 stochastic matrix has at least one steady-state vector. Any such matrix can be written in the form P = 1-α B α 1-B where x and ẞ are constants between 0 and 1. (There are two linearly independent steady-state vectors if a = ß= 0. Otherwise, there is only one.) If P is a stochastic matrix, then a steady-state vector, or equilibrium vector, for P is a probability vector q such that Pq=q Pq=1. 8] Pq=0. Pq=P.
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