91影视

Given that Ais a regular transition matrix of size $n脳n$with equilibrium distribution ${\stackrel{鈫�}{x}}_{equ}$, and$\stackrel{鈫�}{x}$ is any distribution vector in${\mathrm{鈩�}}^n$

Let $i^{th}$element of$\stackrel{鈫�}{x}$ is $x_i$.

This implies$\underset{m鈫�\mathrm{鈭�}}{\lim }\left(A^m\stackrel{鈫�}{x}\right)=\left(\underset{m鈫�\mathrm{鈭�}}{\lim }\right)\stackrel{鈫�}{x}$

Since multiplication of matrix is a linear operator, therefore, it is continuous.

Let A be a regular transition matrix of size$n脳n$.

a. There exists exactly one distribution vector$\stackrel{鈫�}{x}$in ${\mathrm{鈩�}}^n$such that$A\stackrel{鈫�}{x}=\stackrel{鈫�}{x}$. This is equilibrium distribution for A, denoted ${\stackrel{鈫�}{x}}_{equ}$. All the components of ${\stackrel{鈫�}{x}}_{equ}$are positive.

b. If$\stackrel{鈫�}{x}$is any distribution vector in${\mathrm{鈩�}}^n$, then$\underset{m鈫�\mathrm{鈭�}}{\lim }\left(A^m\stackrel{鈫�}{x}\right)={\stackrel{鈫�}{x}}_{equ}$.

c.$\underset{m鈫�\mathrm{鈭�}}{\lim }{A}^m=\left[\begin{array}{ccc}鈫�& & 鈫�\\ {\stackrel{鈫�}{x}}_{equ}& \mathrm{...}& {\stackrel{鈫�}{x}}_{equ}\\ 鈫�& & 鈫�\end{array}\right]$ , which is a matrix whose columns are all ${\stackrel{鈫�}{x}}_{equ}$.

Part C says that$\underset{m鈫�\mathrm{鈭�}}{\lim }{A}^m=\left[\begin{array}{ccc}鈫�& & 鈫�\\ {\stackrel{鈫�}{x}}_{equ}& \mathrm{...}& {\stackrel{鈫�}{x}}_{equ}\\ 鈫�& & 鈫�\end{array}\right]$

So,$\underset{m鈫�\mathrm{鈭�}}{\lim }\left(A^m\stackrel{鈫�}{x}\right)=\left[\begin{array}{ccc}鈫�& & 鈫�\\ {\stackrel{鈫�}{x}}_{equ}& \mathrm{...}& {\stackrel{鈫�}{x}}_{equ}\\ 鈫�& & 鈫�\end{array}\right]\stackrel{鈫�}{x}$

On multiplying the vector and the matrix, the result is a new vector which is a linear combination of the columns of the matrix and the coefficients are the elements of the vector.

So the equation becomes:

$\begin{array}{c}\underset{m鈫�\mathrm{鈭�}}{\lim }\left(A^m\stackrel{鈫�}{x}\right)=x_1{\stackrel{鈫�}{x}}_{equ}+x_2{\stackrel{鈫�}{x}}_{equ}+\mathrm{...}+x_n{\stackrel{鈫�}{x}}_{equ}\\ =(x_1+x_2+\mathrm{...}+x_n){\stackrel{鈫�}{x}}_{equ}\end{array}$

Now, it is given that$\stackrel{鈫�}{x}$ is a distribution vector. So the sum of elements is 1.

Therefore,$\underset{m鈫�\mathrm{鈭�}}{\lim }\left(A^m\stackrel{鈫�}{x}\right)={\stackrel{鈫�}{x}}_{equ}$ , which proves part (b).