1.2.2. Applied Area.

In this section we introduce some applications which will lead to the matrix products of row vector times a column vector, matrix times a column vector and the general matrix times a matrix. These operations are fundamental to many numerical models, and these computations have been carefully implemented on most computers.

Application to Store Inventory. A typical department store will have about 10,000 different items each with a different price per item. The problem is to determine the net worth of the store's inventory. Clearly, we must sum 10,000 numbers where each number is a product of the number of each item and the price of this item. With 10,000 products in the store all this data must be organized in a systematic way. For example, suppose the store has just four types of TVs:

This information maybe recorded in an ordered list of four numbers called a price column vector

P = .

Associated with this order is and inventory column vector

I = where

7 is the number of 27" TVs, 12 is the number of 27" TVs, 3 is the number of 31" TVs and

2 is the number of 50" TVs. The net worth is given in the following form

ITP = 7*100 + 12*400 + 3*900 + 2*2100.

The symbol IT is a row vector associated with the column vector I and is called the transpose of I

IT = [7 12 3 2].

This product is called a row vector times a column vector or dotproduct of two column vectors. Here the net worth is the dotproduct of the inventory vector and the price vector.

The general scheme for n different types of items with item k having price p(k) and i(k) number of item k is as follows:

ITP = i(1)*p(1) + i(2)*p(2) +... + i(n)*p(n) where

I = [i(1) i(2) ... i(n)]T and P = [p(1) p(2) ... p(n)]T are n dimensional column vectors. This product has n multiplications and n-1 additions, and we say it has order n operations. Another order n operation is multiplication of a column vector by a constant. For example, if the above TV store were to triple its inventory, then the new inventory vector would be

3IT = [21 36 9 6].

Matrix operations which have order n operations are classified as BLAS1 which signifies basic linear algebra subroutine of order n to the first power operations. Most computers have BLAS1 subroutines which have been optimized for the best possible performance on that computer.

Application to Investment. An investor has some portions in stocks, bonds, growth stocks and money market. Each year she decides how to reallocate her investments. From past year's reallocations we have

For the remainder of this application we shall keep the order stock, bond, growth and money. The above probabilities can be associated with a row vector to invest in stock

P1T = [.8 .4 .4 .5].

If the current investments are

then by writing this last information in vector form C = [.50 .40 .07 .03]T, we can expect

P1TC = .8*.5 + .4*.4 + .4*.07 + .5*.03

to be the fraction to be invested in stock the next year.

Associated with each investment category is a probability vector

PiT= [p(i,1) p(i,2) p(i,3) p(i,4)] where

p(i,j) is the probability of investing in category i if already invested in category j. Hence, according to the above ordering p(2,4) is the probability of investing in bonds if already invested in growth.

PiTC = the fraction to be invested in category i given the current investments C. We can record this in vector format as

newC = [P1TC P2TC P3TC P4TC]T.

Another way to say the is that the component i of the new investment vector is equal to the dotproduct of the probability vector i and the old investment vector, C.

The probability vectors can be formed into a 4 by 4 matrix or array

P = .

The newC can be viewed as a product of the matrix P and the column vector C where component i of the newC is defined as the product of row i of P times column vector C.

One can repeat this from one year to the next to try to predict future patterns of investment. For example, after two years the prediction will have the form P(PC), after three years P(P(PC) and so forth. We will later examine the behavior of this sequence of investment vectors. This is an example of a Markov chain where P is called a transition matrix from one state to the next state. Each column of P has non negative component whose sum equals one.

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