Matrices (dominant and steady state)

Dec 2014
31
0
Australia
I have been working with dominant matrices as shown below for a game of dice. The diagonal row of zeros indicate that of course the player can not verse themselves, so it is donated as zero e.g T vs T has a 0. The 1's are wins and zero a loss accept for T vs T, J vs J ect (am i correct in assuming so?)
33.png
In the next matrix however (which is an overall dominant matrix squared) you will notice 6 7 6 7 i was just wondering what these represent? is this a diagonally dominate matrix?
Untitled.png
My last question on a different type of matrix is about a steady state matrix. Is there a way of proving22.png
algebraically? I know that if you multiply the steady state by the transition matrix to a large power on n it is the same for any larger power of n+1.

Thank you Giovanni
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey Giovanni55.

When you say Sn does this mean calculating some matrix S^n?

If so you may want to consider diagonalizing it - or using some similar decomposition if you can't do the standard PDP^(-1) since it will be in the form P*D^n*P^(-1) which is a function of n.
 
Dec 2014
31
0
Australia
well another way of putting it is SP^n (roughly equals) = SP^n+1 where s is the steady state matrix and P is the transition (and in this case the route matrix as it is to do with a network) matrix.