Binary, Permutation, ommunication and Dominance Matrices Binary Matrices A binary matrix is a special type of matrix that has only ones and zeros as elements. Some examples of binary matrices; Permutation Matrices A permutation matrix is a square binary matrix where there is only one in each row and column.
An identity matrix is a special permutation matrix. Permutation matrices rearrange the elements in another matrix, however an identity matrix keeps the elements in the other matrix the same. Permutation matrices can be pre or post multiplied. Example Define two matrices into your calculator. $ = & = ' ( ) * +, v Multiply &$. Why does the calculator give this answer? v Multiply $&. What do you notice about the elements in $&? v Multiply $ - &. Notice the elements again. What has happened?
ommunication Matrices Say we look at the workers in a company. Because of their positions and level of power within the company, not every level of employment communicates with the other. For example; v The EO speaks to the General Manager about financial planning v The Assistant Manager speaks to the General Manager about protocol. v The clerk speaks to the General Manager to ask if he wants coffee. v The General manager on any given workday will talk to the EO, the Assistant Manager and the clerk. We can see that EO does not communicate directly with the Assistant Manager on any given work day, but communicates with the General Manager. This is called a one-step communication link.
G A K. = G A K But, the EO can communicate with the Assistant Manager by sending a message via the General Manager, who we can see speaks to both of them. This is called a two-step communication link. While we can use a diagram to work out the one-step and two-step communication links, it is easier to show the two-step links and the total of one-step and two-step links in matrices.. - = G A K 3 G A K
So, if we look at the first row, we can see that the EO can talk to Assistant Manager and lerk through the General Manager represented by (). represents a two-step communication link in matrix. represents a two-step communication link in. -. All of the non-zero elements in the leading diagonal (top to bottom; left to right) of a communication matrix, or its powers, represent redundant links in the matrix. It s not very meaningful to see that EO can talk to himself, so it s not providing new information. This is true for all cases. - = G A K 3 G A K
(Same as above) But we also know that the EO can talk directly to the General Manager so this matrix isn t showing all the possible outcomes. We can find the total number of one and two-step links in a communication matrix system, T, which is evaluated & =. +. - & =. +. - = 3 + G A K G A K
& = 3 Here there are no elements in the matrix, because all of the people can communicate with each other through the help of others. Like before in matrix. -, the leading diagonal in matrix & is redundant. Dominance In sporting competitions, teams and players are said to be dominant. What does that mean? Dominant means being on top, in control, in power etc. Lets say we choose 5 teams from the AFL. We notice that:
v North Melbourne defeated Geelong and Essendon. v Richmond defeated Essendon, North Melbourne and St Kilda. v Essendon defeated Geelong. v Geelong defeated Richmond. v St Kilda defeated North Melbourne, Geelong and Essendon. Which is the dominating team? We could use a network diagram to display this result. Please write down the network diagram for this question, it isn t in the notes J Both Richmond and St Kilda had three wins each so there is a tie. We can find out which is the best team by calculating a dominance score for each team. We do this by constructing a series of dominance matrices. One-step dominances
Lets call the teams by corresponding letters. N= North Melbourne, G= Geelong, E= Essendon, R= Richmond, S= St Kilda. v North Melbourne defeated Geelong and Essendon. v Richmond defeated Essendon, North Melbourne and St Kilda. v Essendon defeated Geelong. v Geelong defeated Richmond. v St Kilda defeated North Melbourne, Geelong and Essendon. = N G E R S N=2 G=3 E= R= S=3 So, what I ve done, indicated in the green, is I ve calculated a one-step dominance score for each team by summing the rows of matrix.
According to this matrix, Geelong and St Kilda both have the same one-step dominance score of 3. Two-step Dominance This is where we see what team has beaten a team who has beaten another team (make sense?). For example, Geelong has a two-step dominance over Richmond because Geelong defeated Essendon who defeated Richmond. We find two-step dominances the same way we find two-step communications. N G E R S - = 2 3 2 N=2 G=6 E= R=3 S=4 Same with communications, with dominance the total score is the one-step matrix plus the two-step matrix.
& = & = + - N G E R S 2 2 3 3 2 3 N=4 G=9 E=2 R=4 S=7 Using these total dominance scores we see that Geelong is the clear winner with 9. St Kilda is second with a score of 7 and North Melbourne and Richmond are equal 3 rd with a score of 4. Please do all F Questions from your textbook J