Linear Algebra and Vector Geometry
Linear Algebra and Vector Geometry MATH 2700
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Eigenvalues Eigenvectors and Differential Equations William Cherry April 2009 The concepts of eigenvalue and eigenvector occur throughout advanced mathematics They are often introduced in an introductory linear algebra class and when introduced there alone it is hard to appreciate their importance The most likely other place one will meet eigenvalues in the undergraduate curriculum is in a course in differential equations although the presence of eigenvalues may be hidden under the surface and not mentioned explicitly The purpose of these notes is to give a quick introduction to the c of and t s in the study of differential equations Only the simplest cases are treated here and a more thorough study is left for a proper course on differential equations 1 A differential equation from calculus Let y ft and suppose that y satis es the differential equation ft Aft which can also be written Ay where A is a constant This equation says that the rate of growth of y is proportional to y and A is the proportionality constant You may remember that such equations are associated to population growth the rate at which a bacteria culture reproduces is proportional to the amount of bacteria already present and radioactive decay the rate at which a substance decays is proportional to the amount of substance there We have chosen to call our constant A to foreshadow the fact that it will be Viewed as an eigenvalue The above equation is called a di erential equation because it is an equation that involves a derivative It is called an ordinary differential equation because it only involves ordinary derivatives and not partial derivatives from multivariable calculus This equation is a rst order equation because it only involves a rst derivative It is also called a linear equation since y and dydt only appear as linear terms and hence the eventual connection with linear algebra Finally the equation is called autonomous because it does not change with time 25 although its solution does Solving a differential equation means nding all possible functions ft that satisfy the equation Unlike most differential equations it happens that the above differential equation is easy to solve and you may have learned how to solve it in your freshman calculus class Namely the variables can be separated so that the left side contains only y and the right side contains only if 7 Adt We can then integrate both sides to get 2 rv v rv and Iva Ir and by actually nding the antiderivatives ln y t c where c is a constant of integration We do not need a constant of integration on each side because the constant on the left can be combined with the constant on the right Now exponentiating both sides of the equation we nd Atc ye ece t Because 50 is also a constant we can denote it simply as C and we have y 05 This is known as the general solution of the differential equation Note that when t 0 then y C and so it is also common to denote C by yo and write 24 2405M lf we specify a value of yo then we have what is called a particular solution to the equation and it is the solution of the differential equation that satis es the initial condition y0 yo 2 Systems of differential equations The easiest way to make a connection to linear algebra is to consider systems of differential equa tions Your textbook discusses examples coming from simple electric circuits but I nd it more fun to consider some Romeo and Juliet examples I learned these examples as a technique for teaching the role of eigenvalues in differential equations from the book Steven H Strogatz Nonlinear Dynamics and Chaos Addison Wesley 1994 The situation is as follows Let R or Rt denote Romeo s affection for Juliet at time t We will say R gt 0 corresponds to positive affection 239e love for Juliet and R lt 0 corresponds to negative affection 239e hate Similarly J will denote Juliet s affection for Romeo at time t Then dRdt and dJdt represent how Romeo and Juliet s affections for each other are changing at an instant in time To begin let s consider a case where Romeo and Juliet s feelings for each other depend only on themselves and not on how the other feels about them More precisely suppose the change in Romeo s affection for Juliet is proportional to how much affection he already has for her and similarly for Juliet ln equations this could be written dR 7R dt 1 N 7J dt 2 Although this is a system of equations the two equations are completely independent of one another Thus they can be solved as above to get R ROeAlt J JOeAZt n u n and Iva u n 3 L However to emphasize the connection with linear algebra let s write the original system in matrix form dRdt i 1 0 R dJdt T 0 A2 J 39 The fact that the matrix is diagonal is what makes the equations so easy to solve lf 1 gt 0 and Romeo starts out with some love for Juliet R0 gt 0 then Romeo s love for Juliet feeds off itself and grows exponentially lf R0 lt 0 then Romeo s hate for Juliet feeds off itself and grows exponentially On the other hand if 1 lt 0 then R decays exponentially so Romeo s passion either positive or negative for Juliet gradually whithers away J uliet s feelings are similarly determined by A2 Although in this case it was not hard to nd an exact formula for Romeo and J uliet s feelings as a function of time often we are not so much interested in such exact formulas and only want to know what happens 239e do Romeo and Juliet fall and eventually remain in love with each other This kind of question can often be answered by drawing what is known as a phase diagram77 or phase portrait77 To plot a phase diagram for our Romeo and Juliet equations at each point B J dRdt dJdt The corresponding phase plane would then look like in the plane we plot the vector By way of example let s take 1 2 and 2 71 Notice for example that at the point 1 1 the vector dRdt 7 2R 7 2 dJdt iJ 71 has been plotted In fact the vectors have been shortened to make the plot more readable because it is mainly the direction of the vector we are interested in and the relative size of the vector compared with its neighboring vectors The actual size of the vector is not important The arrows show how Romeo and J uliet s affections will change We get a solution trajectory by starting at a given initial point and then following the arrows For example suppose that we start at the point B J 051 corresponding to Romeo having a slight interest in Juliet and Juliet having a moderate interest in Romeo Following the arrows we nd the solution trajectory drawn here Thus we see that Juliet s love for Romeo decays over time and Romeo s love for Juliet grows stronger We can see that eventually Romeo will be head over heels in love with Juliet but Juliet will eventually be indifferent toward Romeo As the matrix corresponding to this system was diagonal the two eigendirections are the co ordinate axes Thus we see that no matter where we start because J uliet s affection decays that we end up approaching the eigendirection corresponding to the R axis as an asymptote Of course if we start on the R axis we stay there The J axis is also an eigenspace and if we start on the J axis we stay there but approaching the origin If we start near the J axis we eventually move away from it When this happens the J axis is called a repeller Also we see that eventually we approach the R axis so we call the R axis an attractor 21 A coupled system Now let s look at a more complicated situation This time instead of Romeo and Juliet s feelings feeding off their own feelings for each other suppose that their feelings change based on how the other feels toward them In other words suppose dR a J U 7 bR dt with a and 1 positive constants Thus Romeo grows fonder of Juliet the more she likes him and Juliet grows fonder of Romeo the more he likes her This system can be written in matrix notation lil fHSSllll We will begin by plotting the phase diagram for this system r v r and Iva v n i 5 NN 2 If we add in plots for a few attractor dotted lines We will show the repeller and attractor are the eigendirections of the matrix To nd the eigenvalues of the matrix OS 1 we compute the determinant 7 a b 7A and set it equal to zero to get the characteristic equation so the two eigenvalues are We nd that 6 rv v rv and Iva IT is an eigenvector with eigenvalue v ab and 7 11 1 is an eigenvector with eigenvalue 7v ab These vectors point in the direction of the repeller and attractor Thus we see that depending on where we start either Romeo and Juliet both end up more and more in love or they end up hating each other more and more Who has the stronger feelings is determined by whether a gt b or b gt a The pictures above show the case a gt b In fact we can also use eigenvalues and eigenvectors to solve for R and J in terms of 25 Indeed consider the change of basis matrix l l P Ib T 1ab which has inverse P71 21 b5 a 5 Then 0a 7 M 0 1 lbol Pl 0 will and we therefore have vianew 0 ir dJdt 0 7M J lfwe set m i 1 R dzdt i 1 dRdt L l P l J l then dydt 13 dJdt 7 and so dzdt 7 M 0 m dydt T 0 7M 39 Because this is a diagonal system we know Mt zzoe 7 bt 9905 W To switch back to R and J we use 7 lRlPllW WNW imltmowgt J yogi abt maxTb yoeimt We will not be particularly interested in this explicit formula I only want to point out that by using a change of basis matrix associated to a basis of eigenvectors we can nd an exact formula if we want and that the the eigenvalues appear in these formulas 22 TWO equally cautious lovers Let us now consider the system l Elli l lYi allllv n u n and Iva u n i 7 L where a gt 0 and b gt 0 In this circumstance Romeo and Juliet are both affraid of their own feelings toward the other so the more Romeo loves Julliet the more he pulls back and similarly for Juliet On the other hand they respond positively to the other s affection for them so the more Juliet likes Romeo the more he tends to like her If we nd the characteristic equation for this system we nd which leads to or in other words 7aib We also nd that i J is an eigenvector for the eigenvalue 7ab and that i for the eigenvalue 7a 7 b In order to analyze what happens we need to now consider two cases First consider the case b gt a which corresponds to Romeo and Juliet being more sensitive to each other s feelings than to their own In this case the eigenvalue 7a b is positive and of course 7a 7 b is negative In this case the eigendirection R J will be an attractor and the tranjectories will move out away from the origin so that either Romeo and Juliet both end up loving each other or both end up hating each other is an eigenvector There is one exceptional case Namely if we happen to start on the line B 7 then both Romeo and Juliet s feelings fade to indifference over time and the trajectory ends up at the origin In the case when a gt 1 meaning that Romeo and Juliet are both more sensitive to their own feelings than to each others we nd that both eigenvalues are negative so both Romeo and J uliet s feelings decay toward mutual indifference no matter where they start Because the eigenvalue in the direction R 7 is more negative indicating faster decay in that direction the eigendirection R J remains an attractor andr quot 2 3 An example With imaginary eigenvalues Now consider the system of equations dR a J dB 7 712R dt with a and 1 positive In this circumstance Romeo responds positively to Juliet s affection for him but Juliet likes Romeo more when Romeo dislikes her and conversely she likes Romeo less when Romeo likes her more The phase diagram for this system is 2 AN K In this case we see that a typical solution trajectory looks like NH x c K u u F2 7 n Suppose that things started with Juliet interested in Romeo but Romeo ambivalent to Juliet Then Juliet7s interest in Romeo makes Romeo more fond of Juliet But this then turns Juliet off and she becomes less enthralled with Romeo until she begins to dislike him As Juliet s dislike for Romeo grows Romeo loses his affection for Juliet until nally he also grows to dislike her But this causes Juliet to become more attracted to Romeo until nally Juliet becomes fond of Romeo again and the cycle repeats itself Writing the system that led to this phase portrait in matrix form we get dRdt i 0 a R dJdt 7 7b 0 J 39 This time we nd that the characteristic equation is A2 xiab and so the eigenvalues are pure imaginary iiv ab Since the eigenvalues are imaginary we have no real eigenvectors and so we do not see any asymptotic directions associated to eigendirections in our phase diagram If we allow ourselves to use complex eigenvectors to diagonalize our matrix we will nd the explicit formulas for the solution to this system of equation will involve elk 7quot and e imt Recalling that am coshE25 isinEt and e imquot cosh3t isinOEtt we can re write our solution in terms of cosabt and sinabt Notice that the period of the cycle is directly related to the eigenvalues 24 A spiral To conclude these notes consider the system dRdt i 72 1 R dJdt 7 71 71 J 39 The phase portrait for this system looks like In this case the eigenvalues are complex and contain both a non zero real and imaginary part 3 7 if i 7 2 z 2 The fact that the real part is negative corresponds to exponential decay The imaginary part is related to the period of rotation If the real part had been positive the spirals would move outward as time advances If we were to use complex eigenvectors to diagonalize our matrix we would nd our explicit solution to be built out of 5 3t2 cost2 and 5 3t2 sint2 and again we see the eigenvalue show up in the explicit formula the real part of the eigenvalue affecting the decay rate and the imaginary part of the eigenvalue affecting the period The theme here is that the eigenvalues tell you the qualitative behavior of the system When the eigenvalues are real some of the eigenvectors point in asymptotic directions Other eigenvectors point in the direction that repels solutions away There are many subtleties involved in nding explicit formulas as solutions that have not been touched on here These issues are more properly discussed in a differential equations class In particular I have not addressed here what to do in the case that there is a repeated eigenvalue and the matrix cannot be diagonalized Math 2700 Cherry Key Concepts systems of linear equations and writing them in matrix form augmented matrix Elementary Row Operations Existence and Uniqueness Questions Row Reduction Echelon and Reduced Echelon form Pivot positons Vector notation and equations Homogeneous and Inhomogeneous equations and systems Linear combinations and span Linear independence Connections between span linear independence existence questions uniqueness questions and pivots Linear transformations Testing if a transformation is linear writing a linear transformation as a matrix basic geometric examples rotation dilation shear onetoone also called inj ective and connections to linear independence and pivots onto also called surjective and connections to span of the columns and pivots invertible onetoone and onto also called bijective kernel range or image matrix addition multiplication and transpose How to invert a matrix shortcut for 2 X 2 general procedure for 3 X 3 and larger Invertible Matrix Theorem null space column space determinants calculating by expanding by cofactors calculating by row operations Cramer s Rule Connections between determinant of a linear transformation and volume Vector spaces and subspaces de nition and how to decide if a set is a vector space or subspace o bases and dimension 0 rank 0 relationship between rank dimension of null space dimension of column space and number of columns 0 change of basis and coordinates o Eigenvalues and eigenvectors characteristic polynomial imaginary eigenvalues complex eigenvalues Using eigenvalues and eigenvectors to help understand a linear transformation 0 Connections between eigenvalues eigenvectors and differential equations as in the Romeo and Juliet examples ofquot quot 39 andits quot quotlto39 and o Dot products angle between vectors length of vectors orthogonal o orthogonal projections o orthonormal bases and GramSchmidt orthogonalization 0 least squares problems