definition of Markov matrix and related theorems are showed below 8.4.2Show that the matrix (8.4.21) is a Markov matrix which is not regular. Is A stable? Definition 8.7 Let A = (aij) A satisfies R(n, n) so that aij-0 for i, j = I, . . . , n. If j-1 243 8.4 Markov matrices that is, the components of each row vector in A sum up to 1, then A is called a Markov or stochastic matrix. If there is an integer such that A is a positive matrix, then A is called a regular Markov or regular stochastic matrix. A few immediate consequences follow directly from the definition of a Markov matrix and are stated below. Theorem 8.8 Let A (aij E R(n, n) be a Markov matrix. Then 1 is an eigenvalue of A which enjoys the following properties. (1) The vector(1, ..., 1)' ER" is an eigenvector of A associated to the eigenvalue1 (2) Any eigenvalue λ E C of A satisfies (8.4.2) Theorem 8.9 IfA e R(n, ) is a regular Markov matrix, then the eigenvalue 1 of A is the dominant eigenvalue of A which satisfies the following properties. (1〉 The absolute value of any other eigenvalue λ E C of A s less than 1. That is, Al1 (2) 1 is a simple root of the characteristic polynomial of A 8.4.2Show that the matrix (8.4.21) is a Markov matrix which is not regular. Is A stable? Definition 8.7 Let A = (aij) A satisfies R(n, n) so that aij-0 for i, j = I, . . . , n. If j-1 243 8.4 Markov matrices that is, the components of each row vector in A sum up to 1, then A is called a Markov or stochastic matrix. If there is an integer such that A is a positive matrix, then A is called a regular Markov or regular stochastic matrix. A few immediate consequences follow directly from the definition of a Markov matrix and are stated below. Theorem 8.8 Let A (aij E R(n, n) be a Markov matrix. Then 1 is an eigenvalue of A which enjoys the following properties. (1) The vector(1, ..., 1)' ER" is an eigenvector of A associated to the eigenvalue1 (2) Any eigenvalue λ E C of A satisfies (8.4.2) Theorem 8.9 IfA e R(n, ) is a regular Markov matrix, then the eigenvalue 1 of A is the dominant eigenvalue of A which satisfies the following properties. (1〉 The absolute value of any other eigenvalue λ E C of A s less than 1. That is, Al1 (2) 1 is a simple root of the characteristic polynomial of A