Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for ech component yi of the vector y is liner function of the vector x, tht is, yi i x. It is useful to use mtrix rrys to exmine liner trnsformtions. Consider mtrix A with m rows nd p columns nd second mtrix B with p rows nd n columns. We define the new mtrix C to be the product of A nd B if the ijth element of C is the product of two vectors, the ith row of A nd the jth column of B. An exmple is shown below. Note tht the product of two mtrices A nd B is only defined if the number of columns of A is equl to the number of rows of A. 2 2 4 2 C AB 3, 3 3 3 2 2 C AB 3 Then liner trnsformtion is the product of the mnmtrix A nd the column vector x. y A x Exercise.: Rules of mtrix lgebr () Follow the definition of mtrix multipliction nd estblish the following rules of mtrix lgebr for the cse. (i) A( x z) Ax A z (where nd re prmeters.) (ii) A( B +C) (iii) A( Bx) ( AB ) x AB + AC. (b) The specil mtrix with ones down the digonl nd zeroes elsewhere is the identity mtrix nd written s I. Confirm tht the identity mtrix mps vector onto itself, tht is Ix=x.
Inverse of mtrix Consider the mtrix A 2 2. The ij-th sub-mtrix of A is the mtrix creted by deleting the i-th row nd j-th columns. In the 2 2 cse ech sub-mtrix is just number. They re shown below s second mtrix. m m M 2 2 m2 m 2. Flipping mtrix on its side is clled trnsposing mtrix. The trnsposed mtrix is written s m M m m 2 m 2 Next define the cofctor of the number is odd nd consider the mtrix m to be ( ) i c j m so tht the sign chnges if i j ij ij ij 2 3 c c2 ( ) m ( ) m 2 m m2 2 C 3 4 c2 c ( ) m2 ( ) m m 2 m 2. Define the determinnt of A A 2 2 As is esily checked, A AC = CA A A I. A As long s the determinnt of A is not zero we cn therefore define the inverse mtrix B C A. Then we hve proved tht if A AB=BA=I. We use the nottion A to denote the inverse of A. 2
Then AA A A I. Solution of liner eqution system Consider the liner system of equtions Ax=b where A. Multiply both side by the inverse mtrix A A b A ( Ax) ( A A) x I x x. Exercise.2: Solution of liner eqution system 8 A 7 5 () Solve for the inverse mtrix. (b) Solve the liner eqution system Ax=b if 4 b 3 (c) For generl vector b solve for x(b), the solution to Ax=b. Exercise.3: Liner dependence () Show tht A 2 x x x2 2. (b) If A explin why AC=. (c) Hence show tht 2 c c2 2 Thus the two column vectors of A re linerly relted. (d) Show lso tht the two rows of A re linerly relted. 3
2. Dynmics of liner systems The stte of n economic system is chrcterized by the column vector x t x () t x2() t ( ), t,2.... The initil stte is x () x2() The economy evolves ccording to the following liner system of equtions. x( t ) A x( t), tht is, x ( t ) 2 x ( t) x ( t ) x ( t) 2 2 2 Then x(2) A x(), 2 x(3) Ax(2) A( Ax()) A x(),. x t t ( ) x() A. It is helpful to use spredsheet to compute the sequence x(), x(2),..x(t) for some finite number of periods. Downlod the spredsheet nd you will find tht the mtrix is the one in Exercise 2.: Explosive dynmics? Sheet A ndsheetb depict the evolution of x(t) for T=5 nd T= nd different strting stte x(). () Use SheetC to see how the stte evolves for different initil sttes. (b) Show tht the evolution depicted in one of the first two sheets is unusul. Suggestion: The strting stte for SheetA is on the line. Try strting sttes (i) on this line nd (ii) ner this line. (c) Confirm your conclusions using Sheet2B one which llows choice of how mny periods (up to.) (d) Wht cn you conclude bout the rtio x2( t) / x( t) s t increses for lmost ll initil sttes? 4
Exercise 2.2: x( t ).25 x( t) x ( t ).25 x ( t) 2 2 () Does the stte vry pproximtely s depicted for ll initil x()? (b) Vry the prmeters of the mtrix to see some of the possible outcomes. In prticulr, by incresing or decresing the prmeters long the leding digonl ( nd ) try to chrcterize prmeters tht led the system to dmped oscilltions nd prmeters tht led the system to oscillte explosively. 5
3. Chrcteriztion of the dynmics of liner system (to be discussed in Mth Cmp) To understnd the dynmics we first sk if there is strting point x() such tht the stte simply grows t constnt rte, tht is x( t ) x( t). We cn rewrite this in mtrix form s follows. x t x ( t ) x ( t) ( ) x( t) x2( t ) x2( t) I Since x( t ) A x( t), it follows tht Ax( t) I x( t), tht is B xt ( ) where B A I. If B is invertible xt ( ) B. Thus if we re to find some non zero x () it must be the cse tht B cnnot be inverted. As we hve seen this is the cse if nd only of the determinnt of B is equl to zero. Thus b b b 2 2 2 2 b2 b b2 b ( )( ). Exercise 3. () Consider the mtrix 4 2 A 3. Solve for the two chrcteristic roots (, ). (b) For the first of these chrcteristic roots show tht the stte grows t constnt rte if the x () strting stte is x () x (). Solve lso for the strting stte vector grows t the constnt rte. x x () () x2 () if the stte (c) For ny strting point x () explin why there is unique such tht x() x () ( ) x () (d) Hence show tht x(2) x () ( ) x (). 6
HINT: Remember tht Ax () x () nd Ax () x (). t t (e) Repet the rgument to show tht x( t) x () ( ) x (). (f) Hence explin why, for lrge t, the system grows t pproximtely the rte of the chrcteristic root with the lrgest bsolute vlue. (g) Suppose tht 2 A Use your spred-sheet to depict the evolution of the stte. Solve for the chrcteristic roots if ny. 7
Answers for section Exercise.2: A =- () 2 2 8 A 7 5, 5 7 M 8, 5 8 5 8 M 7 C 7, 5 8 5 8 A 7 7. A It is esy to check the result by multiplying 8 A 7 5 nd 5 8 7 (b) Ax=b. Therefore x x x b 5 8 b - - I A A A 7 b 2 Exercise.3: () 2 x x 2 x2 A x x x x 2 2 2 2 2 x 2 x2 x 2 x2 x x2 x 2 2x x 2 2x x A 2 (b) As shown bove, A AC = CA A A I. A Therefore if A, (c) It follows tht c c 2 2 AC 2 c2 c. 2 c 2 c c2 2 c 2 2 8
Exercise 2. The chrt below shows xt () when T =24 nd the strting stte is very close to the line x x. Note tht for the first 5 periods the stte pproches the origin. But x(t) eventully 5 2 2 strts veering wy. If you choose higher T you will see tht x(t) eventully gets close to the line x2 x. This is 5 true for ll strting sttes not on the line x x. Thus the rtio x2( t) / x( t ) pproches. 2 2 5 Note finlly tht if the strting point is bove the line x x then x () t is lrge nd positive 2 2 for lrge T. If the strting point is below the line then x () t is lrge nd negtive for lrge T. 9
Exercise 2.2 The chrt below shows cse in which the oscilltions re lmost stble. However in this cse the oscilltions re slightly dmped. For higher vlues on the leding digonl the oscilltions become explosive. For lower vlues the cycles become more dmped.