An square matrix is called **regular ** if for some integer all entries of are positive.

**Example**

The matrix

is not a regular matrix, because for all positive integer ,

The matrix

is a regular matrix, because has all positive entries.

It can also be shown that all other eigenvalues of A are less than 1, and algebraic multiplicity of 1 is one.

It can be shown that if is a regular matrix then approaches to a matrix whose columns are all equal to a probability vector which is called the **steady-state ** vector of the regular Markov chain.

where .

It can be shown that for any probability vector when
gets large, approaches to the **steady-state ** vector

.

That is

where .

It can also be shown that the steady-state vector q is the only vector such that

Note that this shows q is an eigenvector of A and is eigenvalue of A.