10.6 Show that if for some m x m matrix P, Pa for every m x 1 vector a, then P is doubly stochastic. 10.6 Show that if for some m x m matrix P, Pa for every m x 1 vector a, then P is doubly s...
2. A Markov chain is said to be doubly stochastic if both the rows and columns of the transition matrix sum to 1. Assume that the state space is {0, 1,....m}, and that the Markov chain is doubly stochastic and irreducible. Determine the stationary distribution T. (Hint: there are two approaches. One is to solve T P and ( 1 in general for doubly stochastic matrices. The other is to first solve a few examples, then make an educated guess...
2. A Markov chain is said to be doubly stochastic if both the rows and columns of the transition matrix sum to 1. Assume that the state space is {0, 1,....m}, and that the Markov chain is doubly stochastic and irreducible. Determine the stationary distribution T. (Hint: there are two approaches. One is to solve T P and ( 1 in general for doubly stochastic matrices. The other is to first solve a few examples, then make an educated guess...
36. (a) Every 2 X 2 stochastic matrix has at least one steady state vector. (b) If A and B are stochastic matrices, then so is ½ (A + B).
Theory: A vector with nonnegative entries is called a probability vector if the sum of its entries is 1. A square matrix is called right stochastic matrix if its rows are probability vectors; a square matrix is called a left stochastic matrix if its columns are probability vectors; and a square matrix is called a doubly stochastic matrix if both the rows and the columns are probability vectors. **Write a MATLAB function function [S1,S2,P]=stochastic(A) which accepts a square matrix A...
(a) Suppose we want to solve the linear vector-matrix equation Ax b for the vector x. Show that the Gauss elimination algorithm may be written bAbm,B where m 1, This process produces a matrix equation of the form Ux = g , in which matrix U is an upper-triangular matrix. Show that the solution vector x may be obtained by a back-substitution algorithm, in the form Jekel (b) Iterative methods for solving Ax-b work by splitting matrix A into two...
Let X be an irreducible and aperiodic Markov chain with m <, and suppose that the transition matrix is doubly stochastic. Show that mt it is the limit distribution. Let X be an irreducible and aperiodic Markov chain with m
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
If A is an m × n matrix, and x is an n × 1 vector, then the linear transformation y = Ar maps R" to R", so the linear transformation should have a condition number, condar (x). Assume that |I-ll is a subordinate norm. a. Show that we can define condar (x)-|All llrI/IAxll for every x 0. b. Find the condition number of the linear transformation at[ 2] using the oo-norm. c. Show that condAr(x) IIA for all x....
1) Show that for every 1 Sisn, P(AA)>o 2) Show that PA, n nA")-P(AJPA,İA, )PA,İA, n As) P(A"IA, n nA"-.). Remark. This identity is called the compound probability theorem and is for instance useful in situations where the pašt has an influence on the future (and is in some sense the probabilistic version of the "multiplicative rule") 3) (Application) Consider an urn with 6 identical blue balls and 4 identical red balls. Take one after the other 3 balls at...