Download Matrices and Markov chains

Matrices and Matrix Multiplication
De¯nition: An m £ n matrix A is a rectangular array of numbers with m rows and n columns. The
number in the ith row at the j th column is denoted as Aij : For example, if
0
2
A=@ 5
3
1
1
2 A
1
then A11 = 2; A12 = 1; A21 = 5; A22 = 2; A31 = 3; and A32 = 1:
If A is an m £ n matrix and B is an n £ p matrix, then the product A£B, also written as AB, is the
m £ p matrix whose element in the ith row and j th column is Ai1 B1j + Ai2 B2j + ¢ ¢ ¢ + Ain Bnj : If the
number of columns of A is not the same as the number of rows of B, then the product AB is not de¯ned.
Matrix multiplication is an example of multiplication that is not commutative. Even if AB and BA
are both de¯ned, it is not true in general that AB=BA. Multiplication is associative, however: If
(AB)C=A(BC) provided all the products listed are de¯ned.
You get the element in the product C = AB in the ith row and jth column of C by placing your left
index ¯nger at the beginning of the ith row of A and your right index ¯nger at the top of the jth column
of B and step by step moving the left ¯nger across and right ¯nger down, multiplying and summing as
you proceed.
Checking
your work
0
1with Excel:
0 Suppose1
1 2 4
1 0
A= @ 2 3 2 A and B= @ -1 1 A
-1 5 1
3 1
To ¯nd the product AB using Excel, you would enter the matrix A starting at cell A1, for example.
Suppose you enter the matrix B starting at F1, for example. Since A in a 3 £ 3 matrix and B is a 3 £ 2
matrix, the product will be a 3 £ 2 matrix. Suppose we want the product to appear starting at cell D5.
Click on cell D5 and then click on f¤ , then Math & Trig, then choose MMULT. (Matrix multiplication).
The ¯rst array is from A1:C3. The second array is from F1:G3. After ENTER, you won't see all the
product yet. You will only see the ¯rst entry. Highlight from D5 to E7, since the product will be a 3 £
2 matrix. Now here is the strange part: First hit the F2 key (usually at the top of the keyboard.) Then
push Ctrl, Shift, and Enter all at the same time. The product should now appear. Your screen should
be something like this:
Markov Chains:
The basic idea is that when probability of a certain event depends only on the immediately preceding
observations, you can form a tree, or chain, to predict future probabilities.
Notation: Recall that p(AjB) means the probability of an event A happening if you know the event B
happened. For example, suppose B is the event that two cards were drawn from a full deck of cards and
that the two cards were red cards. Now let A be that a card drawn from a deck of cards is red. Since we
assume B happened, there are 50 cards left, 26 black cards and 24 red cards left. So p(AjB) = 24=50:
Example (This should be of particular interest to those majoring in advertising:)
DUZ is the name of a laundry detergent. The probability that the next purchase of a laundry detergent
will be DUZ given that the current purchase is DUZ is 0.7. Hence the probability that the next purchase
of a laundry detergent will not be DUZ given that the current purchase is DUZ is 0.3. Suppose we also
know that the probability that the next purchase of a laundry detergent will be DUZ given that the
current purchase is not DUZ is 0.4, hence the probability that the next purchase of a laundry detergent
will not be DUZ given that the current purchase is not DUZ if 0.6. Summary up to this point:
p(next purchase is DUZ j current purchase is DUZ)= 0.7
p(next purchase is not DUZ j current purchase is DUZ) = 0.3
p(next purchase is DUZ j current purchase is not DUZ) = 0.4
p(next purchase is not DUZ j current purchase is not DUZ) = 0.6
Let D stand for the purchase if DUZ, and D' stand for the purchase is not DUZ.
Notation: Form a transition matrix T:
0
µ D D ¶
:7 :3
T =
:4 :6
D
D0
Suppose currently a market analysis shows that the probability of a random shopper buying a DUZ
detergent is .25 and a di®erent brand is .75
Form a current probability matrix P:
P =
D D0
(:25 :75)
Form a tree to get the probabilities that the next purchase will be DUZ or not DUZ, and then form the
matrix product PT :