The Estimation for the Eigenvalues of Stochastic Matrices

In the past two decades, due to study on matrix theory and some engineering background problems, many scholars dedicated to special matrix , and obtained some important and valuable results (TingZhu, 2007 Yigeng Huang, 1994). But in combination matrix theory, combinatorics, probability theory (especially Markov chain), mathematical economics and reliability theory etc. areas there is a special class of non-negative stochastic matrix, which in recent years becomes concerned. This article discusses location, distribution and estimate of the eigenvalue for stochastic matrix . Section 2 introduces the concepts of stochastic matrix and generalized stochastic matrix. Section 3 gives a few estimation theorems of stochastic matrix eigenvalue, also the eigenvalue distribution for tensor product of two stochastic matrices is obtained. In section 4, we discuss the eigenvalue distribution for generalized stochastic matrices.


Introduction
In the past two decades, due to study on matrix theory and some engineering background problems, many scholars dedicated to special matrix , and obtained some important and valuable results (TingZhu, 2007-Yigeng Huang, 1994).But in combination matrix theory, combinatorics, probability theory (especially Markov chain), mathematical economics and reliability theory etc. areas there is a special class of non-negative stochastic matrix, which in recent years becomes concerned.This article discusses location, distribution and estimate of the eigenvalue for stochastic matrix .Section 2 introduces the concepts of stochastic matrix and generalized stochastic matrix.Section 3 gives a few estimation theorems of stochastic matrix eigenvalue, also the eigenvalue distribution for tensor product of two stochastic matrices is obtained.In section 4, we discuss the eigenvalue distribution for generalized stochastic matrices.

Basic concept
Definition 1.If the sum of elements in every row in the n order non-negative matrix A is 1, A is row stochastic matrix; If the sum of elements in every column in the n order non-negative matrix A is 1, A is column stochastic matrix; If both A and A T are row stochastic matrices, A is double stochastic matrix; Row stochastic matrix, column stochastic matrix and double stochastic matrix are called stochastic matrix, denoted by S (n).
Definition 2. If the sum of elements in every row in the n order non-negative matrix A is s, A is called the first generalized row stochastic matrix; If the sum of elements in every column in the n order non-negative matrix A is s, A is called the first generalized column stochastic matrix; If both A and A T are the first generalized row stochastic matrices, A is called the first generalized double stochastic matrix; The first generalized row stochastic matrix, the first generalized column stochastic matrix and the first generalized double stochastic matrix are called the first generalized stochastic matrix, denoted by S I (n).Definition 3. If the absolute value sum of elements in every row in the n order matrix A is 1, A is called the second generalized row stochastic matrix; If the sum of elements in every column in the n order matrix A is 1, A is called the second generalized column stochastic matrix; If both A and A T are the second generalized row stochastic matrices, A is called the second generalized double stochastic matrix; The second generalized row stochastic matrix, the second generalized column stochastic matrix and the second generalized double stochastic matrix are called the second generalized stochastic matrix, denoted by S II (n).
Definition 4. If the absolute value sum of elements in every row in the n order matrix A is s, A is called the third generalized row stochastic matrix; If the sum of elements in every column in the n order matrix A is s, A is called the third generalized column stochastic matrix; If both A and A T are the third generalized row stochastic matrices, A is called the third generalized double stochastic matrix; The third generalized row stochastic matrix, the third generalized column stochastic matrix and the third generalized double stochastic matrix are called the third generalized stochastic matrix, denoted by S III (n).
S I (n), S II (n) and S III (n) are called generalized stochastic matrices.Obviously, for S (n), S I (n), S II (n) and S III (n), we have the following simple conclusions: (1).S (n)

Eigenvalue estimate of stochastic matrix
Theorem 1. Suppose A = (a i j ) n×n is a row stochastic matrix and where λ(A) is denoted the whole eigenvalues of matrix A, G(A) is Gerschgorin disc of matrix A.
Proof: Since λ is an arbitrary eigenvalue of matrix A = (a i j ) n×n and X and and from AX = λX, we get Since λ is an arbitrary eigenvalue of matrix A = (a i j ) n×n , then we have Theorem 2. Suppose A = (a i j ) n×n is a row stochastic matrix and where λ(A) is denoted the whole eigenvalues of matrix A, G(A) is denoted disc whose center is T r(A) n and radius is ).
Proof: From paper (Yixi Gu, 1994) and for arbitrary matrix A, we have And because A = (a i j Similarly, we get where λ(A ⊗ B) is denoted the whole eigenvalues of tensor product for matrix A and matrix B, G(A ⊗ B) is the oval region of the product for elements of Gerschgorin disc whose center is m 1 = min{a ii , i = 1, 2, • • • , n} and radius is 1 − m 1 and Gerschgorin disc whose center is m 2 = min{b j j , j = 1, 2, • • • , m} and radius is 1 − m 2 .
From theorem 1, we have Therefore, the eigenvalues of tensor product for matrix A and matrix B are located in the oval region G(A ⊗ B).
Theorem 4. Suppose A = (a i j ) n×n is a row stochastic matrix and where λ(A) is denoted the whole eigenvalues of matrix A, G(A) is denoted generalized Gerschgorin disc of matrix A.
Proof: Because λ is an arbitrary eigenvalue of matrix A = (a i j ) n×n and X The theorem is proven.
Multiply right each item of the above equation with y * m , then λt m y m y * m = t m a mm y m y * m |a m j | = 1 − a mm .Therefore, |λ − m| = |λ − a mm + a mm − m| ≤ |λ − a mm | + |a mm − m| ≤ 1 − a mm + a mm − m = 1 − m.