LS.4 Decoupling Systems - MIT Mathematics

1 Decoupling Systems1. Changing common way of handling mathematical models of scientific orengineering problems isto look for a change of coordinates or a change of variables which simplifies the problem. Wehandled some types of first-order ODE s the Bernouilli equation and the homogeneousequation, for instance by making a change of dependent variable which converted theminto equations we already knew how to solve. Another examplewould be the use of polaror spherical coordinates when a problem has a center of example from physics is the description of the acceleration of a particle moving inthe plane: to get insight into the acceleration vector, a newcoordinate system is introducedwhose basis vectors aretandn(the unit tangent and normal to the motion), with the resultthatF=mabecomes simpler to are going to do something like that here. Starting with a homogeneous linear systemwith constant coefficients, we want to make a linear change of coordinates which simplifiesthe system .

2 We will work withn= 2, though what we say will be true forn >2 would a simple system look? The simplest system is one with a diagonal matrix:written first in matrix form and then in equation form, it is(1)(uv) =( 100 2) (uv),oru = 1uv = you can see, if the coefficient matrix has only diagonal entries, the resulting system really consists of a set of first-order ODE s, side-by-side as it were, each involving only itsown variable. Such a system is said to bedecoupledsince the variables do not interact witheach other; each variable can be solved for independently, without knowing anything aboutthe others. Thus, solving the system on the right of (1) gives(2)u=c1e 1tv=c2e 2t,oru=c1(10)e 1t+c2(01)e we start with a 2 2 homogeneous system with constant coefficients,(3)x =Ax,and we want to introduce new dependent variablesuandv, related toxandyby a linearchange of coordinates, , one of the form (we write it three ways):(4)u=Dx,(uv)=(a bc d) (xy),u=ax+byv=cx+ callDthedecoupling matrix.

3 After the change of variables, we want the system tobe decoupled, , to look like the system (1). What should we choose asD?The matrixDwill define the new variablesuandvin terms of the old in order to substitute into the system (3), it is really the inverse toDthat we need;we shall denote it NOTES: LS. LINEAR Systems (5)u=Dx,x=Eu,E=D the Decoupling , we first produceE; thenDis calculated as its inverse. We needboth matrices:Dto define the new variables,Eto do the are now going to assume that the ODE systemx =Axhastwo real and distincteigenvalues; with their associated eigenvectors, they are denoted as usual in these notes by(6) 1, ~ 1=(a1b1); 2, ~ 2=(a2b2).The idea now is the following. Since these eigenvectors are somehow special to thesystem, let us choose the new coordinates so that the eigenvectors become the unit vectorsiandjin theuv- system . To do this, we make the eigenvectors the two columnsof thematrixE; that is, we make the change of coordinates(7)(xy)=(a1a2b1b2) (uv),E=(a1a2b1b2).

4 With this choice for the matrixE,i=(10)andj=(01)in theuv- system correspond in thexy- system respectively to the first and second columnsofE, as you can see from (7).We now have to show that this change to theuv- system decouples the ODE systemx =Ax. This rests on the following very important equation connecting a matrixA, oneof its eigenvalues , and a corresponding eigenvector~ :(8)A ~ = ~ ,which follows immediately from the equation used to calculate the eigenvector:(A I)~ = 0 A ~ = ( I)~ = (I ~ ) = ~ .The equation (8) is often used as the definition of eigenvector and eigenvalue: aneigenvector ofAis a vector which changes by some scalar factor when multipliedbyA; the factor is theeigenvalueassociated with the it stands, (8) deals with only one eigenvector at a time. Werecast it into the standardform in which it deals with both eigenvectors simultaneously. Namely, (8) says thatA(a1b1)= 1(a1b1),A(a2b2)= 2(a2b2).These two equations can be combined into the single matrix equation(9)A(a1a2b1b2)=(a1a2b1b2) ( 100 2),orA E=E( 100 2),as is easily checked.

5 Note that the diagonal matrix of s must be placed on the right inorder to multiply thecolumnsby the s; if we had placed it on the left, it would havemultiplied therowsby the s, which is not what we Decoupling SYSTEMS19 From this point on, the rest is easy. We want to show that the change of variablesx=Eudecouples the systemx =Ax, whereEis defined by (7). We have,substitutingx=Euinto the system , the successive equationsx =AxEu =A EuEu =E( 100 2)u,by (9);multiplying both sides on the left byD=E 1then shows the system is decoupled:u =( 100 2) a matrixAwith two real and distinct eigenvalues, the matrixEin (7)whose columns are the eigenvectors ofAis called aneigenvector matrixforA, and thematrixD=E 1is called thedecoupling matrixfor the systemx =Ax; the newvariablesu, vin (7) are called thecanonical can alter the matrices by switching the columns, or multiplying a columnby a non-zero scalar, with a corresponding alteration in thenew variables; apartfrom that, they are the systemx =x yy = 2x+ 4ymake a linear change of coordinates which decouples the system ; verify by direct substitutionthat the system becomes matrix form the system isx =Ax,whereA=(1 12 4).

6 We calculate firstE, as defined by (7); for this we need the eigenvectors. The charac-teristic polynomial ofAis 2 5 + 6 = ( 2)( 3) ;the eigenvalues and corresponding eigenvectors are, by theusual calculation, 1= 2, ~ 1=(1 1); 2= 3, ~ 2=(1 2).The matrixEhas the eigenvectors as its columns; thenD=E 1. We get (cf. , (2)to calculate the inverse matrix toE)E=(1 1 1 2),D=(2 1 1 1).By (4), the new variables are defined byu=Dx,u= 2x+yv= x y . NOTES: LS. LINEAR SYSTEMSTo substitute these into the system and check they they decouple we usex=Eu,x=u+vy= u these into the original system (on the left below) gives us the pair of equationson the right:x =x yy = 2x+ 4yu +v = 2u+ 3v u 2v = 2u 6v;adding the equations eliminatesu; multiplying the top equation by 2 and adding eliminatesv, giving the systemu = 2uv = 3vwhich shows that in the new coordinates the system is work up to this point assumes thatn= 2 and the eigenvalues are real and if this is not so?

7 If the eigenvalues are complex, the corresponding eigenvectors will also be complex, ,have complex components. All of the above remains formally true, provided we allow all thematrices to have complex entries. This means the new variablesuandvwill be expressedin terms ofxandyusing complex coefficients, and the decoupled system will have some branches of science and engineering, this is all perfectly acceptable, and one getsin this way a complex Decoupling . If one insists on using realvariables only, a Decoupling isnot there is only one (repeated) eigenvalue, there are two cases, as discussed in . In thecomplete case, there are two independent eigenvalues, but as pointed out there ( ), the system will be be automatically decoupled, be a diagonal matrix. Inthe incomplete case, there is only one eigenvector, and Decoupling is impossible (since inthe decoupled system , bothiandjwould be eigenvectors).Forn 3, real Decoupling requires us to findnlinearly independent real eigenvectors, toform the columns of the nonsingular matrixE.

8 This is possible ifa) all the eigenvalues are real and distinct, orb) all the eigenvalues are real, and each repeated eigenvalue is the end of , we note again the important theorem in linear algebra whichguarantees Decoupling is the matrixAis real and symmetric, ,AT=A, all its eigenvalues will bereal and complete, so that the systemx =Axcan always be : Section Ordinary Differential Notes and Exercisesc Arthur Mattuck and 1988, 1992, 1996, 2003, 2007, 20111