Math 307 Spring, 2005 Hentzel Time: 10:00 to 10:40 MWF Room: 205 Carver Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://orion.math.iastate.edu/hentzel/class.307.05 Text: Linear Algebra and its Applications, Third Edition David C. Lay Friday, February 11 2.6 Assignment Page 156 Problems 1,2,3,4,9 Main Idea: Manipulating a can of worms. Key Words: Leontief, input-output Goal: Understand the Leontief input-output model and see how to set it up and use it. Previous Assignment Page 150 Problem 24 (QR Factorization) Suppose that A = QR where Q and R are nxn, R is invertible and upper triangular, and Q T has the property that Q Q = I. Show that for n each B in R , the equation AX = B has a unique solution. What computations with Q and R will produce the solution. AX = B QR X = B T RX = Q B Now back solve to get X. ----------------------------------------- Page 150 Problem 25 (Singular Value Decomposition) Suppose that T A = UDV , where U and V are nxn matrices with T T the property that U U = I and V V = I, and where D is a diagonal matrix with positive numbers o ,o , ..., o on the diagonal. Show that 1 2 n -1 A is invertible, and find a formula for A . -1 -1 T A = V D U -1 where D has the inverses of the elements of D on its diagonal. -------------------------------------------- Page 150 Problem 26 (Spectral Factorization) Suppose a 3x3 matrix -1 A admits a factorization as A = P D P , where P is some invertible 3x3 matrix and D is the diagonal matrix. | 1 0 0 | D = | 0 1/2 0 | | 0 0 1/3 | Show that this factorization is useful when computing high powers of A. Find a fairly 2 3 k simple formulas for A , A , and A (k a positive integer, using P and the entries of D. | 1 0 0 | n | n | -1 A = P | 0 (1/2) 0 | P | n | | 0 0 (1/3) | =============================================================================== Now for the Leontief Open model. We are studying the internal workings of a country. We study the manufacturing of goods, the production of food, and the production of energy like dams and hydro electric plants. The interaction between the sectors is very very hard to understand. For example, during the petroleum shortage back in the 1970 the airlines and the trucking industry was badly hurt and suffered many layoffs. What happened to the truck repair industry? Strangely, they had a boom time. Since the trucks were idle anyway, this was a good time to get the needed work done. The prime example of this tinkering with the economy came with Russia and the five year plans where they tried to plan the economy to meet projected goals. The same was true in the USA during the war. We had to make sure that what was produced could actually reach the front. It was pointless to concentrate of producing tanks if there were no ships to deliver them, and it was pointless to make ships if there was nothing to transport. The idea was to divide the resources so that there were no bottle necks and every things flowed smoothly. Now days, one can use the idea to work the stock market. When you read in the paper that there is a new oil embargo, or that there is a ban on DDT, or what ever, the big boys have already had that information and the stocks market prices have already been adjusted. What is left is for one to get in on the second tier. Those are the industries which are not so obviously affected. You can buy the data on the US market and use that to predict what will happen if the current supplies are adjusted. The Leontief method works with an intact economy. We simply do the same things we always did, but do more or less of them. We do not introduce new radical ways of doing things. For example, before World War Two, air planes were made a dozen at a time. If a part did not fit, the mechanic would get out his file and adjust it. Then he would do the next part and the next part after that. In the auto industry, if a part did not fit, they went back and changed the die for making that part. During the war, the airplanes were made in the car plants. This was not just a scaling up of the way things were, but was a new type of operation. You have an economy that is working. The following table tells ======================= how the goods produced by sectors R,S,T are interrelated. We only use three sectors. The real economy uses about 40 sectors. R S T Consumer Total R 50 20 40 70 180 S 20 30 20 90 160 T 30 20 20 50 120 The total produced by sector R was 180 units. Of this, the consumer got 70, the rest was used up by the other industries. In this data the columns are what is consumed. Think of the Consumer column. This tells how much of R,S,T are eaten by the consumers. The R column tells you how much Plant R eats. The COLUMN says that in the process of making 180 units, the plant R used: 50 units of its own output, 20 units of the S output, and 30 units of the T output. To make 1 unit of R, Plant R uses 50/180 of R, 20/180 of S, and 30/180 of T. Simply divide the COLUMN under R by the total R output. Simply divide the COLUMN under S by the total S output. Simply divide the COLUMN under T by the total T output. | 50/180 20/160 40/120| A = | 20/180 30/160 20/120| | 30/180 20/160 20/120| To make x units of R, and y units of S and z units of T requires internal consumption of | 50/180 20/160 40/120| |x| | 20/180 30/160 20/120| |y| | 30/180 20/160 20/120| |z| | 60 | Now, suppose that we wish to provide | 110 | for the customers. | 60 | | x | What values of production | y | should be ordered so that after | z | | 60 | internal consumption, | 110 | remains for the customers. | 60 | | x | Of the total output of | y | some has to be used as internal | z | consumption and the rest goes to demand. We write AX+D = X. The left hand side is the internal consumption + what the consumers use. The right hand side is the total produced. In matrix form: | 50/180 20/160 40/120| |x| | 60 | | x | | 20/180 30/160 20/120| |y| + |110 | = | y | | 30/180 20/160 20/120| |z| | 60 | | z | the new demand To solve the system AX+D=X for the new demand vector one does these steps. AX+D=X D = X-AX D = IX-AX D = (I-A)X -1 (I-A) D = X. Leontief Problem. The production of the plants R, S, and T for some period of time is given below. R S T Consumer Total R 10 10 30 30 80 S 10 20 10 20 60 T 60 20 10 10 100 The Leontief input-output model for the open model is: X = AX + D. (1) What is the matrix A? | 145 | (2) Solve the equation X = AX+D for X when D = | 290 | | 145 |