Proving Matrix A's Entries are Between 0 & 1

[tom08]

Homework Statement

Hi, everyone! I encounter a problem as follows:

I have got a matrix A, all the entries in A is between 0 and 1. and the sum of each row of A is 1.

Can we say that all the entries in A^k is also between 0 and 1 ?

Can everyone kindly show me how to prove it when answer is yes :-)

 

[Dick]

Can you show that if a_i and b_i are lists of numbers (like rows or columns of a matrix) and sum(a_i)=1 then min(b_i)<=sum(a_i*b_i)<=max(b_i)?

 

[Altabeh]

Homework Statement

Hi, everyone! I encounter a problem as follows:

I have got a matrix A, all the entries in A is between 0 and 1. and the sum of each row of A is 1.

Can we say that all the entries in A^k is also between 0 and 1 ?

Can everyone kindly show me how to prove it when answer is yes :-)

The answer is YES! To show how it happens, the only hint is that you start by setting two of each row-entries equal to 1/2 in a variety of ways where you will get the maximum numbers after taking the matrix to the power 2 and the larger the power is, the smaller components get. If you proved the theorem for 1/2, it would be obviously proven for 1/n.

AB