Linear Algebra

284 Chapter Three. Maps Between Spaceseach game. We have this.⎛⎞ ⎛ ⎞ ⎛ ⎞0 0 0 0 . . . p 0-0 (n) p 0-0 (n + 1)p 0 0 0 . . .p 1-0 (n)p 1-0 (n + 1)1 − p 0 0 0 . . .p 0-1 (n)p 0-1 (n + 1)0 p 0 0 . . .p 2-0 (n)=p 2-0 (n + 1)0 1 − p p 0 . . .p 1-1 (n)p 1-1 (n + 1)⎜⎝ 0 0 1 − p 0 . . . ⎟ ⎜⎠ ⎝p 0-2 (n) ⎟ ⎜⎠ ⎝p 0-2 (n + 1) ⎟⎠....An especially interesting special case is when the teams are evenly matched,p = 0.50. This table below lists the resulting components of the n = 0 throughn = 7 vectors. (The code to generate this table in the computer algebra systemOctave follows the exercises.)Note that evenly-matched teams are likely to have a long series — there is aprobability of 0.625 that the series goes at least six games...0 − 01 − 00 − 12 − 01 − 10 − 23 − 02 − 11 − 20 − 34 − 03 − 12 − 21 − 30 − 44 − 13 − 22 − 31 − 44 − 23 − 32 − 44 − 33 − 4n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 71 0 0 0 0 0 0 00 0.5 0 0 0 0 0 00 0.5 0 0 0 0 0 00 0 0.25 0 0 0 0 00 0 0.5 0 0 0 0 00 0 0.25 0 0 0 0 00 0 0 0.125 0 0 0 00 0 0 0.375 0 0 0 00 0 0 0.375 0 0 0 00 0 0 0.125 0 0 0 00 0 0 0 0.0625 0.0625 0.0625 0.06250 0 0 0 0.25 0 0 00 0 0 0 0.375 0 0 00 0 0 0 0.25 0 0 00 0 0 0 0.0625 0.0625 0.0625 0.06250 0 0 0 0 0.125 0.125 0.1250 0 0 0 0 0.3125 0 00 0 0 0 0 0.3125 0 00 0 0 0 0 0.125 0.125 0.1250 0 0 0 0 0 0.15625 0.156250 0 0 0 0 0 0.3125 00 0 0 0 0 0 0.15625 0.156250 0 0 0 0 0 0 0.156250 0 0 0 0 0 0 0.15625Markov chains are a widely-used applications of matrix operations. Theyalso give us an example of the use of matrices where we do not consider thesignificance of the maps represented by the matrices. For more on Markov chains,there are many sources such as [Kemeny & Snell] and [Iosifescu].