What is a Regular Transition Matrix

[ha9981]

I have to learn a section from my textbook and I can't seem to undertand what a regular transition matrix is. The definition given is: A transition matrix is regular if some integer power of it has all positive entries. Now an identity matrix isn't regular, but I am pretty sure all integer powers of it have positive entries. I mean no transition matrix I have seen so far is negative except for (1-P) matrices.


P =

0.2 0.1 0.7
0.6 0.4 0.2
0.2 0.5 0.1

for this matrix, is it regular because all values in it are positive and so all integer powers of P will remain positive? Do I have to mention that all rows add up to 1, is that important in being a regular matrix?

also, how would u test to prove a transition matrix isn't regular?

 

[HallsofIvy]

No, any power of the identity matrix does NOT have positive entries. Any power of the identity matrix is the identity matrix and all entries off the main diagonal are 0, not positive.

Are you confusing "positive" with "non-negative"? A "transition" matrix always has entries between 0 and 1 (and every column sums to 1) so any power of a transition matrix has non-negative entries. But not necessarily positive entries.

 

[ha9981]

I might have confused the idea. It said that a identity matrix alternates with powers. Is there somewhere I can learn this online as I am clearly having trouble understanding from my text.

 

[HallsofIvy]

The n by n identity matrix has the property that IA= AI= A for any n by n matrix A (that's the whole point of "identity"). In particular, taking A= I, I²= I(I)= I, I³= I(I²)= I(I)= I, etc.

I have no idea what you mean by "an identity matrix alternates with powers".

 

[ha9981]

A transition matrix is regular if some integer power of it has all positive enteries. What does this mean. Can anyone give me examples of non regular as my text shows matrices with positive enteries and asks to show that it isn't regular, I don't see how that it is possible for that transition matrix to be non regular.

 

[StellaWolf]

A matrix is called positive if all of it's entries are positive. A square matrix is called regular if some power of it is positive.

Example:

A =

3 1
2 2

B =

3 1
2 0

Matrix A is positive, Matrix B is not (as it contains 0). However B is regular, since:

B^2 =

11 3
6 2

is positive.

So to prove a matrix is NOT regular you need to show that no matter how many times you multiply a matrix by itself the result will never be positive (contain all positive values, 0 is not positive).