4 Homogeneous Constant Matri Systems, Part II Let us now epand our discussions begun in the previous chapter, and consider homogeneous constant matri systems whose matrices either have comple eigenvalues or have incomplete sets of eigenvectors 4 Solutions Corresponding to Comple Eigenvalues Eigenpairs of Real Matrices Remember that the real and imaginary parts of a comple number z are those real numbers and y, respectively, such that z + iy, and that the comple conjugate z of z is z iy All these notions etend to matrices having comple components in the obvious manner: Given a matri M whose entries are comple, we define the comple conjugate M of M to be the matri formed by replacing each entry of M with that entry s comple conjugate, and the real and imaginary parts of M are simply the corresponding matrices of the real and imaginary parts of the components of that matri It is easy to show that the standard identities for the conjugates of comple numbers such as ( z ) z, (az) (a )(z ) and z w z w also hold for analogous epression involving matrices and column vectors In particular, for any comple value r, (Au) (A )(u ) and (ru) r u But if (as we ve been assuming) the components of A are real numbers, then a a a N a a a N A a a a N a a a N A, a N a N a N N a N a N a N N /5/4
Chapter & Page: 4 Homogeneous Constant Matri Systems, Part II and thus, Au ru (Au) (ru) (A )(u ) r u A(u ) r u, giving us the first part of the following lemma: Lemma 4 Let A be a real N N constant matri Then (r, u) is an eigenpair for A if and only if (r, u ) is an eigenpair for A Moreover, if r is not real, then the real and imaginary parts of u form a linearly independent set The proof of the second part the claimed linear independence is not difficult, but may distract us from our core narrative So we ll place it in an appendi for later verification (see lemma 4 in section 46) The Corresponding Real-Valued Solutions Let s now look at the solutions to A when the N N matri A has comple eigenvalues, still assuming that A is a real constant matri From lemma 4, we know that the comple eigenpairs occur in comple conjugate pairs; so let (r, u) and (r, u ) be such a pair, with and u r λ + iω, r λ iω, a + ib a b a + ib a + i b a + ib a N + ib N u u u N a N u a ib b N So one fundamental pair of solutions to A corresponding to the comple conjugate pair of eigenvalues λ + iω and λ iω is (t) ue rt [a + ibe (λ+iω)t and (t) u e r t [a ibe (λ iω)t, which we could rewrite in terms of sines and cosines since e (λ±iω)t e λt e ±iωt e λt [cos(ωt) ± i sin(ωt) Recall that we had a similar situation arise when solving single homogeneous linear differential equations with constant coefficients (see chapters 6 and 8) And here, just as there, we prefer to have solutions that do not involve comple values So let s try adapting what we did there to get a corresponding fundamental pair of solutions to A in terms of real-valued functions only We start by splitting our two solutions into their real and imaginary parts: [a ± ibe (λ±iω)t [a ± ibe λt [cos(ωt) ± i sin(ωt) e λt [a cos(ωt) b sin(ωt) ± ie λt [b cos(ωt) + a sin(ωt)
Solutions Corresponding to Comple Eigenvalues Chapter & Page: 4 Letting R (t) the real part of (t) e λt [a cos(ωt) b sin(ωt) and I (t) the imaginary part of (t) e λt [b cos(ωt) + a sin(ωt), we can write our two solutions to A as (t) R (t) + i I (t) and (t) R (t) i I (t) In turn, we can easily solve this pair of equations for R and I, obtaining (4a) (4b) R (t) (t) + (t) and I (t) i (t) i (t) Being linear combinations of solutions to the homogeneous linear system A, R and I must also be solutions to this system of differential equations Note that both are in terms of real-valued functions only And it is a fairly easy eercise (using the second part of lemma 4) to show that { R, I} is a linearly independent pair It is this pair that we will use as the pair of real-valued solutions corresponding to eigenvalues λ ± iω Theorem 4 Let A be a real constant N N matri Then the comple eigenvalues of A occur as comple conjugate pairs Moreover, if u is an eigenvector corresponding to eigenvalue λ + iω (with ω ), then a corresponding linearly independent pair of real-valued solutions to A is given by { R (t), I (t) } where and R (t) the real part of (t), I (t) the imaginary part of (t) (t) ue (λ+iω)t ue λt [cos(ωt) + i sin(ωt) By the way, memorizing formulas (4a) and (4b) for R (t) and I (t) is not necessary, or even advisable for most people Instead, just epand ue (λ+iω)t into its real and imaginary parts, and use those parts for R (t) and I (t)! Eample 4: Let s solve the system (t) (t) 4 y z (t) z Setting up the characteristic equation: det[a ri r det 4 r r ( 4 r) [ ( r) + 9 (r + 4) [ r + 4r +
Chapter & Page: 4 4 Homogeneous Constant Matri Systems, Part II Hence, either which you can easily solve, obtaining r + 4 or r + 4r +, r 4, r + i and r i as the eigenvalues for our matri By now, you should have no problem is showing that u [,, T is an eigenvector corresponding to eigenvalue r 4, giving us (t) e 4t as one solution to our system For the pair of solutions corresponding to the conjugate pair of eigenvalues r + i and r i, we first need to find an eigenvector u corresponding to r +i Letting u [ α,β, γ T, we see that [A r I u α 4 ( + i) β γ ( + i) α 4 ( + i) β ( + i) γ i α 6 i β i γ So α, β and γ must satisfy the algebraic system iα + β + γ α (6 + i)β + γ α + β iγ The second equation in this system clearly tells us that β The first and third equations are equivalent (if this is not obvious, multiply the third by i ) So α and β must satisfy iα + β + γ,
Solutions Corresponding to Comple Eigenvalues Chapter & Page: 4 5 which can be rewritten as γ iα Thus, all the eigenvectors corresponding to eigenvalue + i are given by α α u β α γ iα i where α is arbitrary Picking α, we have the single eigenvector u i So, a single solution corresponding to to the conjugate pair of eigenvalues r + i and r i is (t) ue r t e (+i)t e t [cos(t) + i sin(t) i i To find the corresponding pair of real-valued solutions, we first epand out the last formula for (t), (t) e t [cos(t) + i sin(t) i cos(t) + i sin(t) e t i cos(t) sin(t) cos(t) sin(t) cos(t) sin(t) e t + i e t + i e t sin(t) cos(t) sin(t) cos(t) Taking the real and imaginary parts, we get cos(t) sin(t) R (t) e t and I (t) e t sin(t) cos(t) So, we now have a set { (t), R (t), I (t) } cos(t) sin(t) e 4t, e t, e t sin(t) cos(t) of solutions to our homogeneous linear system of differential equations It should be clear that this set is linearly independent (if its not clear, compute the Wronskian at t ) Hence, it is a fundamental set of solutions to our system, and a general solution is given by (t) c e 4t cos(t) + c sin(t) e t + c sin(t) cos(t) e t
Chapter & Page: 4 6 Homogeneous Constant Matri Systems, Part II 4 Two-Dimensional Phase Portraits with Comple Eigenvalues Let us now see what patterns are formed by the trajectories of a system A when A has comple eigenvalues λ + iω and λ iω with ω As before, we ll let (t) (t) y(t) The General Solutions In the previous section, we saw that, if u a+ib is any eigenvector corresponding to eigenvalue λ + iω, then { R, I } is a linearly independent pair of solutions to A where R (t) e λt [a cos(ωt) b sin(ωt) and I (t) e λt [b cos(ωt) + a sin(ωt) Hence, (t) c R (t) + c I (t) (4) is a general solution for our system A However, for no apparent reason, let s also consider C r (t t ) where C and t are two real constants Using some basic trigonometric identities, you can easily confirm that C R (t t ) Ce λ[t t [ a cos(ω[t t ) b sin(ω[t t ) Ce λt cos(ωt )e λt [a cos(ωt) b sin(ωt) + Ce λt sin(ωt )e λt [b cos(ωt) + a sin(ωt) c R (t) + c I (t) where with c ρ cos(θ) and c ρ sin(θ) ρ Ce λt and θ ωt So, C r (t t ) a solution of A for any pair of real constants C and t Moreover, if we are first given real values c and c, we can then find real constants C and t so that C R (t t ) c R (t) + c I (t) by simply taking the polar coordinates (ρ,θ) of the point with Cartesian coordinates (c, c ) and setting t θ ω and then C ρe λt
Two-Dimensional Phase Portraits with Comple Eigenvalues Chapter & Page: 4 7 This means that we have two general solutions at our disposal Theorem 4 Let A be a real matri with comple eigenvalues and let r λ + iω and r λ iω with ω, R (t) e λt [a cos(ωt) b sin(ωt) and I (t) e λt [b cos(ωt) + a sin(ωt) where a and b are, respectively, the real part and the imaginary part of an eigenvector corresponding to r Then two general solutions for A are c R (t) + c I (t) and C R (t t ) where c, c, C and t are arbitrary (real) constants It immediately follows that, for the A s begin considered now, all the trajectories of A are simply scaled versions of the trajectory of R (t) Consequently, if we can get a reasonable sketch of one nonzero trajectory, all the others in our phase portrait can be obtained by just scaling the one already sketched We will use this in constructing the phase portraits that follow There are three cases to consider: λ, λ > and λ < Trajectories when λ If λ, then e λt and R (t) a cos(ωt) b sin(ωt) From a simple observation, R( t + π ω ) ( [ a cos ω t + π ) ( [ b sin ω t + π ) ω ω a cos(ωt + π) b sin(ωt + π) a cos(ωt) b sin(ωt) R (t), we know the components of R (t) call them R (t) and y R (t) are periodic with a period of π / ω That is, ( R (t), y R (t)) repeatedly goes through the same points on the plane as t increases by increments of π / ω With a little thought, you will realize that this means these trajectories are closed loops Moreover, by the same sort of calculations just done, you can easily verify that R( t + π ) R (t) for all t, ω telling us that, for each point on the trajectory of R, there is another point on the trajectory with the origin right in the middle of these two points So, not only is this trajectory a closed loop, this loop must be centered about the origin In fact, it can be shown that this loop is an ellipse centered about the origin (trust the author on this, or turn to section 46 where it is proven)
Chapter & Page: 4 8 Homogeneous Constant Matri Systems, Part II As a result of all this, sketching a phase portrait is fairly easy First sketch a minimal direction field Then sketch an elliptic trajectory about the origin using the direction field to get the general shape and the direction of travel Finally, scale this trajectory to get a sequence of concentric elliptic trajectories Include the dot at the origin for the critical point, and you have a phase portrait for this system Note that we don t actually need the eigenvector u a+ ib to sketch a phase portrait And the only reason you need the eigenvalue r iω is to verify that it is imaginary! Eample 4: ou can easily verify that A has eigenvalues r ± ±i with corresponding eigenvectors u ± a ± ib ± i where and So general solutions to our system are given by (t) c R (t) + c I (t) and (t) C R (t t ) R (t) the real part of u + e it I (t) the imaginary part of u + e it cos(t) cos(t) + sin(t) sin(t) Since the eigenvalues are imaginary, we know the trajectories are ellipses centered about the origin To get a better idea of the shape of these ellipses and the direction of travel along these ellipses, we ll compute at a few well-chosen points using A : At (, ) : At (, ) : At (, ) : At (, ) : Then we sketch corresponding direction arrows at these points, and at other points using the symmetries in the direction field discussed in section 94 That gives the minimal direction field sketched in figure 4a Sketching an ellipse that best matches this direction field then gives the trajectory also sketched in figure 4a Alternatively, you can plot R (t) for a few well-chosen values of t, and get the direction of travel from the derivative of R (t) at some convenient value of t
Two-Dimensional Phase Portraits with Comple Eigenvalues Chapter & Page: 4 9 (a) (b) Figure 4: (a) A minimal direction field with one matching elliptic trajectory, and (b) a phase portrait of the system in eample 4 Scaling this one ellipse by various constants, adding the dot for the critical point and erasing the direction field then gives the phase portrait in figure 4b for our system In practice, we may not always need to sketch the the trajectory for R (t) as carefully as we did in the above eample Often, it is sufficient to simply note that the trajectories are all concentric ellipses centered at the origin and to determine the direction of travel on each by simply finding the direction arrow at one point In this case, the critical point (, ) is not called a node or saddle point For obvious reasons, we call it a center Observe that the corresponding equilibrium solution is stable, but not asymptotically stable, since the trajectory of any solution (t) with (t ) for some t is an ellipse right around (, ) By the way, when sketching a trajectory for a cos(ωt) b sin(ωt), it is tempting to suspect that the vectors a and b directly correspond to the two aes of the elliptic trajectory Well, in general, they don t To see this, just contemplate the last eample and figure 4a (Still, the lines of the aes are easily determined see eercise 46 on page 4 ) Trajectories when λ > If λ > then all the trajectories are given by scaling the single trajectory of R (t) e λt [a cos(ωt) b sin(ωt) As we just saw, the [a cos(ωt) b sin(ωt) factor in this formula for R (t) repeatedly traces out an ellipse as t varies Multiplying by e λt which increases to + as t + and decreases to as t converts this ellipse into a spiral that spirals outward as t + and spirals inward to (, ) as t, wrapping around the origin infinitely many times in either case Whether the trajectories are spiralling out in the clockwise or counterclockwise direction
Chapter & Page: 4 Homogeneous Constant Matri Systems, Part II (a) (b) Figure 4: (a) A minimal direction field with one matching spiral trajectory, and (b) a phase portrait of the system in eample 4 in which λ > can be determined by finding the direction of travel indicated by computed at a convenient point via A As usual, a minimal direction field may aid your sketching! Eample 4: Let us just consider sketching a phase portrait for A when A [ As you can easily verify, A has eigenvalues r ± i The fact that these eigenvalues are comple with positive real parts tells us that the trajectories of A are spirals with the direction of travel being away from the origin Computing at a few well-chosen points: At (, ) : At (, ) : At (, ) : At (, ) : [ [ [ [ 5 5 From these vectors and the symmetries discussed in section 94, we get the minimal direction field sketched in figure 4a, which also includes a spiral trajectory matching that direction field Note that this trajectory is spiralling outwards in a clockwise direction Adding more spirals matching this minimal direction field (and erasing the direction field) then gives us the phase portrait in figure 4b
Second Solutions When the Set of Eigenvectors is Incomplete Chapter & Page: 4 Figure 4: A phase portrait for A for a case where A has comple eigenvalues with negative real parts For these systems, the origin classified as a spiral point (for obvious reasons), and is clearly unstable Trajectories when λ < The only significant difference between this case and the previous is that e λt decreases to as t + and increases to + as t All this does is to change the direction of travel We still get spiral trajectories, but with the direction of travel on each being towards the origin The result is a phase portrait similar to that sketched in figure 4 The origin is still called a spiral point, but you can clearly see that the equilibrium solution (t) is asymptotically stable 4 Second Solutions When the Set of Eigenvectors is Incomplete Up to now, we ve assumed that our matri A has a complete set of eigenvectors That is, we ve assumed that, for each eigenvalue r the algebraic multiplicity of r the geometric multiplicity of r where (just in case you forgot) the algebraic multiplicity of r while the multiplicity of r as a root of the characteristic polynomial, det[a r I, the geometric multiplicity of r the number of eigenvectors in any basis for the eigenspace of r
Chapter & Page: 4 Homogeneous Constant Matri Systems, Part II Now, let s suppose that an eigenvalue r has algebraic multiplicity of two or more, but geometric multiplicity one, with u being a corresponding eigenvector That gives us one solution (t) ue rt to A It should seem reasonable to epect that there is another solution corresponding to this eigenvalue other than a constant multiple of Let s look for this second solution Based on what we know about second solutions to homogeneous linear differential equations with constant coefficients (from chapter 6), we might suspect that a second solution can be generated by just multiplying the first by t, (t) t (t) tue rt Alas, it doesn t work Try it yourself to see what happens ou get a leftover term that does not cancel out But the attempt may lead you to try adding something to deal with that leftover term So let us try (t) [tu + we rt where w is a yet unknown vector: d dt (t) A d ( ) [tu + we rt A[tu + we rt dt [u + tru + rwe rt [tru + Awe rt u + tru + rw tru + Aw u + rw Aw Aw rw u And since Aw rw Aw riw [A riw, we can rewrite our last equation as [A riw u (4) This is what w must satisfy! Eample 44: Let us consider the system A with A Solving for the eigenvalues, det[a ri det r r (r ),
Second Solutions When the Set of Eigenvectors is Incomplete Chapter & Page: 4 we get a single eigenvalue r with algebraic multiplicity Any eigenvector u [α,β T then must satisfy [A Iu [ [ α β β, which tells us that β and α is arbitrary So any eigenvector of this matri can be written as α u α Taking α gives us the single eigenvector u Thus, r is an eigenvalue with algebraic multiplicity but geometric multiplicity One solution to A is then given by (t) ue rt e t According to our derivation above, another solution is given by (t) [tu + we t where w satisfies [A Iw u To find such a w, plug w [α,β T into the last equation, and solve for α and β : ( [A I w u ) α β β Hence, β /, α is arbitrary, and w [ α [ + α That is, w is [, / T plus any constant multiple of the one eigenvector u For simplicity, let us take that constant multiple α to be α Then, for our second solution, we have (t) [tu + we rt ( t + [ ) e t [ t e t
Chapter & Page: 4 4 Homogeneous Constant Matri Systems, Part II Clearly, is not a constant multiple of Hence, our system A has { } t e t, e t as a fundamental pair of solutions, and ( ) (t) c [ e t + c [ t e t c [ + c [ t e t as a general solution And if, for esthetic reasons, we don t like fractions eplicitly appearing in our general solutions, we can replace c with C, rewriting the above as (t) c [ e t + c [ t e t ( c [ ) t + C e t The key to our finding a second solution lay in finding a w satisfying [A riw u when (r, u) is an eigenpair for A ou may have some concern about this being possible After all, since r is an eigenvalue, det[a ri, telling us that the matri A ri is not invertible So what guarantee is there that the above key equation has a solution w? Well, for matrices there is the following lemma, which you ll get to verify yourself in eercise 4 on page 4 Lemma 44 Let A be a matri with eigenpair (r, u) If r has algebraic multiplicity two, but geometric multiplicity of only one, then there is a w, which is not a constant multiple of u, such that [A riw u Do note that the w in the above lemma is not unique After all, if then, for any constant c, simply because [A riw u for some w, [A riw u with w w + cu, [A ri(w + cu) [A riw + c[a riu u + c Back to differential equations: As an immediate corollary of the above lemma and the work done before the previous eample, we have Much more general versions of this lemma will be discussed in section 45
Two-Dimensional Phase Portraits with Incomplete Sets of Eigenvectors Chapter & Page: 4 5 Theorem 45 Let A be an matri with eigenpair (r, u) If r has algebraic multiplicity two, but geometric multiplicity of only one, then the general solution to A is given by where w is any vector satisfying (t) c ue rt + c [tu + we rt [A riw u ou may wonder if we can epand this discussion to find third and fourth solutions when the algebraic multiplicity is greater than two The answer is yes, but let s look at the trajectories of the solutions we ve found when A is a simple matri before discussing this epansion 44 Two-Dimensional Phase Portraits with Incomplete Sets of Eigenvectors If A is a matri with a single eigenvalue r having algebraic multiplicity two but geometric multiplicity one, then, as we just saw, a general solution to A is given by (t) c ue rt + c [tu + we rt where u is an eigenvector corresponding to eigenvalue r, and w satisfies [A riw u The trajectories of this when c are simply the straight-line trajectories corresponding to the eigenpair (r, u), and, along with the critical point, are the first trajectories to sketch in our phase portrait To simplify our sketching of the trajectories when c, let s rewrite the above formula for as (t) c [ (αu + w)e rt + tue rt where α c / c Note that () c (αu + w) This may help us make rough sketches of the trajectories To help find the direction of travel and any tangents, we can use either the derivative, d dt c [ [(rα + )u + rw)e rt + tue rt, or, to remove any distracting scaling effects, we can use any positive multiple of this Let us use [ T(t) e rt d (rα + )u + rw c + t t dt t t ru Note that while lim T(t) c [ + ( )ru c ru, t lim T(t) c [ + (+)ru +c ru t + This tells us that, if t + is a very large positive value and t is very large negative value, then the tangents to the trajectories at these two values of t
Chapter & Page: 4 6 Homogeneous Constant Matri Systems, Part II but are nearly parallel to the eigenvector u, are in directly opposite directions Trajectories when r > If r > and c, then and lim (t) t lim c [ (αu + w)e rt + tue rt t lim (t) lim c [(αu + w) + tu e rt t t So each trajectory appears to start at (, ) and continues infinitely beyond the origin as t goes from to + This, alone, tells us that the direction of travel on each nonequilibrium trajectory is away from the origin Now, in our general comments, we saw that, if (t) is a solution with c, then and () c (αu + w) lim T(t) ±c ru t ± In particular, if c >, then the trajectory of (t) appears to start at the origin, tangent to u but directed in the direction opposite to u As t increases to t, (t) traces a path that bends in the direction of w to pass through the point corresponding to c (αu + w) As t continues to get ever larger, this path continues to bend and slowly flattens out until the direction of travel is in nearly the same direction as u If c <, then the trajectory of (t) is just a mirror reflection of the trajectory of a corresponding solution having positive c The end result is a phase portrait similar to that sketched in figure 44a This is a phase portrait for the system A with A from eample 44 In that eample, we found that A only had r as an eigenvalue, and that all eigenvectors were constant multiples of u [, T We also obtained w [, / T Do note, however, that you don t really need to know w to sketch this phase portrait ou can (as we ve done several times before) start by sketching the straight-line trajectories for the solutions ±ue rt, then add a minimal direction field and use this direction field to sketch a number of other nonequilibrium trajectories Just keep in mind that these trajectories start at the origin tangent to u and bend around to ultimately be headed in the opposite direction And don t forget the dot at the origin for the equilibrium solution In these cases, the equilibrium solution (t) is clearly an unstable equilibrium The critical point (, ) is classified as a node Those who wish to be more precise refer to it as an improper node
Incomplete Sets of Eigenvectors Chapter & Page: 4 7 (a) (b) Figure 44: Trajectories for a system A when all the eigenvectors of A are parallel to the ais assuming (a) the eigenvalue is positive, and (b) the eigenvalue is negative By the way, we ve not sketched enough of the trajectories in figure 44a to really see that the tangents become parallel to the eigenvector as t gets large That is because the magnitude of (t) increases like e rt, much faster than T(t) approaches ±c ru Consequently, in practice, if you use a scale large enough to see the tangents being almost parallel to the eigenvector at large values of t, then the behavior of the trajectories near the origin becomes almost invisible Trajectories when r < The analysis just done assuming r > can be redone with r < The only real difference is that the directions of travel will be reversed With r <, the direction of travel will be towards the origin The result will be a phase portrait similar to that in figure 44b The origin is a node (again, improper ), and the equilibrium solution (t) is now an asymptotically stable equilibrium Trajectories when r This is left as an eercise (eercise 4 on page 4 ) 45 Complete Sets of Solutions Corresponding to Incomplete Sets of Eigenvectors It is relatively easy to etend the ideas presented in section 4 for finding a second solution corresponding to an eigenvalue with geometric multiplicity one but algebraic multiplicity of two Well, it s relatively easy least as long as the geometric multiplicity remains one and we know enough linear algebra So let us start with that case
Chapter & Page: 4 8 Homogeneous Constant Matri Systems, Part II When the Geometric Multiplicity Is One To begin, here are two lemmas that, together, appropriately generalize lemma 44: Lemma 46 Let (r, u) be an eigenpair for some N N matri A, and let m be the algebraic multiplicity of r If the geometric multiplicity of r is only one, then there is a linearly independent set of m vectors { w, w,, w m } such that (a) w u, and (b) if m > then [A riw k w k for k,,, m However, there is no v such that [A riv w m Lemma 47 Let {r, r,, r J } be the complete set of distinct eigenvalues for an N N matri A, and suppose that the geometric multiplicity of each eigenvalue is one For each r j, let m j be the algebraic multiplicity of r j, and let { } w j,, w j,,, w j,m j be any set of vectors such that W j w j, is an eigenvector for A corresponding to eigenvalue r j, and if m j > then [A riw j,k w jk for k,,, m j Then the union of the W j s is a linearly independent set of N vectors The w k s and w j,k s in the above lemmas are sometimes referred to as generalized eigenvectors for the matri Unfortunately, the arguments outlined in eercise 4 for proving lemma 44 do not readily etend to the more general cases considered by the above lemmas To verify these lemmas, we really should delve deeper into the general theory of linear algebra and further develop such concepts as generalized eigenvectors and the Jordan form of a matri - concepts rarely mentioned in elementary linear algebra courses But that would take us far outside the appropriate bounds of this tet So either trust the authora or (better yet) take a more advanced course in linear algebra and learn how to prove the above lemmas yourself Back to differential equations: Let A be an N N matri with eigenpair (r, u), and assume r has algebraic multiplicity m greater than one, but geometric multiplicity of only one By the arguments already given in section 4, it immediately follows that, if the w k s are as given in the first lemma above, then the first two solutions corresponding to eigenvalue r are (t) w e rt and (t) [ tw + w e rt (44a)
Incomplete Sets of Eigenvectors Chapter & Page: 4 9 where w u, and w is any column vector statisfying [A ri w w (44b) And if the algebraic multiplicity of r is greater than two, then you can easily etend the computations done in section 4 to show that a third solution is given by where w is any column vector satisfying (t) [ t w + tw + w e rt (44c) [A ri w w (44d) (see eercise 4) And once you ve done that (and noted that k () w k ensures the linear independence of the set of k s ), the road to generating a linearly independent set of m solutions to A should be clear, at least when r has algebraic multiplicity m and geometric multiplicity one Traveling that road is left to the interested reader But we can at least do a simple eample using the above formulas! Eample 45: Consider the system A where 4 A 4 4 It is easily verified that det(a ri) (4 r) So this matri has one eigenvalue r 4, and this eigenvalue has algebraic multiplicity three It is also easily seen that all eigenvectors are constant multiples of u So the geometric multiplicity of r 4 is one From this and our last lemma it follows that we have a linearly independent set { w, w, w } with w u, and with w and w being any vectors satisfying [A 4Iw w and [A 4Iw w Letting w [α,β,γ T, we see that [A 4Iw w is simply α β, γ yielding α is arbitrary, β and γ
Chapter & Page: 4 Homogeneous Constant Matri Systems, Part II Taking α for convenience, we have w Now consider [A 4Iw w Letting w [α,β,γ T, this becomes α β, γ which reduces to the easily solved system β + γ and γ From this, it immediately follows that α is arbitrary, β 9 and γ Again, we might as well take α, giving us w 9 9 Applying the formulas given in set (44) for the corresponding solutions to the system A, we have (t) w e rt e 4t, (t) [ tw + w e rt t + e 4t t e 4t and (t) [ t w + tw + w e rt t + t + e 4t t 6t e 4t 9 9 So, a general solution to our system A is (t) c (t) + c (t) + c (t) c e 4t + c t e 4t + c t 6t e 4t 9 t t C + C + C 6t e 4t
Incomplete Sets of Eigenvectors Chapter & Page: 4 When the Geometric Multiplicity Is Greater Than One If the geometric multiplicity of an eigenvalue is two or more, then we must account for the fact that the corresponding basic set of eigenvectors contains two or more eigenvectors This complicates things a bit, especially since the eigenvectors first found might not be quite the ones generating the other generalized eigenvectors ou can see that in the following generalization of lemma 46: Theorem 48 Assume r is an eigenvalue for some N N matri A, and let m and µ be, respectively, the algebraic and geometric multiplicities of r Then N m µ, and there is a unique collection of µ integers {γ,γ, γ µ }, along with corresponding linearly independent sets of vectors W r,k { w k,, w k,,, w k,γ k } for k,,,µ such that: γ + γ + + γ µ m The set {w,, w,,,w µ, } is a linearly independent set of eigenvectors for A corresponding to eigenvalue r The set of all w k, j s is linearly independent 4 For k,,,µ: (a) If γ k >, then [A riw k, j w k, j for j,,γ k (b) There is no v such that [A riv w k,γ k While the above theorem assures us that the collection of γ k s is unique for each eigenvalue r of A, it does not assure us that the corresponding sets of w k, j s are unique In fact, they are not For convenience, let us refer to any set of vectors W r as a complete set of generalized eigenvectors for an N N matri A corresponding to eigenvalue r if and only if W r can be given as W r { w k,, w k,,, w k,γ k : k,,,µ } where: The value µ is the geometric multiplicity of eigenvalue r The set {w,, w,,,w µ, } is a linearly independent set of eigenvectors for A corresponding to eigenvalue r For k,,,µ: (a) If [A riv w k, has no solution, then γ k Otherwise γ k > (b) If γ k >, then [A riw k, j w k, j for j,,γ k (c) There is no v such that [A riv w k,γ k
Chapter & Page: 4 Homogeneous Constant Matri Systems, Part II In practice, a complete set of generalized eigenvectors corresponding to an eigenvalue r can be found starting with the eigenvectors originally found, and requiring that the w k, j s satisfy the three conditions above (with the last being used to find the γ k s ) The previous theorem assures us that such sets eist The one below, generalizing lemma 47, assures us that any process of generating complete sets of generalized functions will generate all the generalized eigenvectors needed for our purposes Theorem 49 Let {r, r,, r J } be the set of all distinct eigenvalues for an N N matri A, and, for j,,, J, let Then: W j a complete set of generalized eigenvectors corresponding to r j Each W j is a linearly independent set of m j vectors with m j being the algebraic multiplicity of r j The set of all the vectors in all these W j s is a linearly independent set of N vectors These two theorems can either be developed using lemmas 46 and 47, or developed independently, and the lemmas viewed as corollaries of the theorems Alternatively, all the lemmas and theorems in this section can be viewed as corollaries of a theorem we will discuss (but not prove) later (theorem 4 on page 4 5) In any case, you either must trust the author on these theorems, or take a suitable course on linear algebra When the situation is as described in the two theorems, then each set of w k, j s described in theorem 48, { w k,, w k,,, w k,γ k }, generates a corresponding linearly independent set of solutions to A, { k,, k,,, k,γ k }, following the scheme just discussed for the case where the geometric multiplicity is one, k, (t) w k, e rt, k, (t) [ tw k, + w k, e rt, k, (t) [ t w k, + tw k, + w k, e rt, One major difference, however, is that you do not know the initial eigenvector w k, All you know is that it must be some linear combination of the µ eigenvectors u, and u µ originally found corresponding to the eigenvalue being used So finding a suitable linear combination w k, a u + a u + + a µ u µ is part of solving [A riw k, w k, for each k
Incomplete Sets of Eigenvectors Chapter & Page: 4 In addition, you do not initially know the γ k s So you just have to iteratively solve for the w k, j s until you reach the point where [A riw k, j k, j w has no solution w k, j ou then know γ k j for that k and j! Eample 46: Consider the system A where 4 A 4 4 It is easily verified that det(a ri) (4 r) So this matri has one eigenvalue r 4, and this eigenvalue has algebraic multiplicity three It is also easily seen that all eigenvectors are linear combinations of the two eigenvectors u and u So the geometric multiplicity of r 4 is two From this and our last lemma it follows that we have a linearly independent set { w,, w,, w,} with w, and w, being eigenvectors of A, and with [A 4I w, w, We also know the corresponding fundamental set of solutions to A is {,,,,, } where, (t) w, e 4t,, (t) w, e 4t and, (t) [ tw, + w, e 4t Now, consider the equation [A 4I w, w, Since w, must be an eigenvector, it must be a linear combination of the two eigenvectors already obtained; that is, σ w, σ u + τu σ + τ τ
Chapter & Page: 4 4 Homogeneous Constant Matri Systems, Part II for some, yet unknown, pair of constants σ and τ Letting w, [α,β,γ T, we now have [A 4I w, w, α σ β γ τ β σ β τ Consequently, α, β, γ, σ and τ can be any values satisfying β σ τ Taking σ, and α γ then gives us σ α w, u + u and w, β τ γ For w, we can take any linear combination of u and u such that {w,, w,, w, } is linearly independent An obvious choice is w, u u Finally, then, we can write out the general solution to A as (t) c, (t) + c, (t) + c, (t) where and, (t) w, e 4t e 4t,, (t) w, e 4t e 4t, (t) [ tw, + w, t e 4t t + e 4t e 4t t That is, (t) t c + c + c e 4t t Finally, we should note that the eigenvalue in question may be comple, r λ + iω If A is real (as we ve been assuming), then all of the above still holds, and, following the discussion at the start of this chapter, we can find a complete set of real-valued solutions to A corresponding to r and r by taking the real and imaginary parts of the comple-valued solutions generated by the procedures just discussed
Appendi for Sections 4 and 4 Chapter & Page: 4 5 46 Appendi for Sections 4 and 4 In this section, we will verify two claims made in sections 4 and 4; namely, The claim of the linear independence of of the real and imaginary parts of a comple eigenvector made in lemma 4 on page 4 The claim made on page 4 7 that the trajectory of R is an ellipse about the origin Linear Independence of Two Vectors Here is the claim and the proof: Lemma 4 Let (r, u) be an eigenpair for a real N N matri A, and assume r is not real Then the real and imaginary parts of u form a linearly independent set PROOF: Let λ and ω be, respectively, the real and imaginary parts of r, and let a and b be the real and imaginary parts of u, so that, and u a + ib and u a ib, r r (λ + iω) (λ iω) iω Since we are assuming r is not real, we must have r r (45) For the moment, suppose {a, b} is not linearly independent This would mean that either b or that a is a constant multiple of b, a γ b If b, then On the other hand, if a γ b, then u a u u a + ib (γ + i)b and u a ib (γ i)b From fact that eigenvectors are nonzero we know that u, u and γ ± i are nonzero Hence, we would have u γ + i γ i u Either way, if {a, b} is not linearly independent, then u must be a constant nonzero multiple of u, and, thus, must be an eigenvector corresponding to r as well as r But then, (r r )u ru r u Au Au, which, since u, tells us that r r,
Chapter & Page: 4 6 Homogeneous Constant Matri Systems, Part II contrary to inequality (45), which followed immediately from the basic assumption that r is not real So it is not possible to have both that r is not real and {a, b} is not linearly independent If r is not real, then it cannot be true that {a, b} is not linearly independent; that is, if r is not real, {a, b} must be linearly independent Ellipticity of Trajectories On page 4 7, it was claimed that the trajectory of R (t) a cos(ωt) b sin(ωt) is an ellipse when (iω, u) is an eigenpair and a and b are the real and imaginary parts of u Since, as we just saw, {a, b} is linearly independent (and, hence, so is {a, b} ), this claim will follow immediately from the following lemma Lemma 4 Let (t) [(t), y(t) T be given by (t) v cos(ωt) + w sin(ωt) where ω is a nonzero real number and {v, w} is a linearly independent pair of vectors whose components are real numbers Then the path traced out by ((t), y(t)) on the Cartesian plane as t varies is an ellipse centered at the origin Our approach to verifying this lemma will be to verify that the and y components of (t) satisfy an equation for such an ellipse So let us start with little review (and possibly an etension) of what you know about equations for ellipses Coordinate Equations for Ellipses Recall the canonical equation for an ellipse centered at the origin and with its aes parallel to the coordinate aes is α + y β where α and β are nonzero real numbers With the same assumptions on α and β, α y β and α + y β are equations for symmetric pairs of hyperbolas about the origin, and the graph of α y β is empty (ie, contains no points) since no (, y) can satisfy this equation Since these are the only possibilities, we have
Appendi for Sections 4 and 4 Chapter & Page: 4 7 -ais y y-ais -ais θ -ais Figure 45: An -coordinate system obtained by rotating the y-coordinate system by an angle θ Lemma 4 Let a and c be real numbers, and assume the polynomial equation has a nonempty graph Then: a + cy If ac >, the graph is an ellipse centered at the origin If ac <, the graph is a pair of hyperbolas Now assume we have a nonempty graph of a polynomial equation of the form A + By + Cy (46) where A, B and C are real numbers To deal with this, we invoke a new -coordinate system obtained by rotating the original y-coordinate system by an angle θ as indicated in figure 45 If you check, you will find that the (, y) and (, ) coordinates of any single point are related by y [ cos(θ) sin(θ) sin(θ) cos(θ) [ (47) Using this, we can rewrite our polynomial equation (equation (46)) in terms of and, obtaining a + b + c (48) where a, b and c are formulas involving cos(θ) and sin(θ) Now, the clever choice for θ is the one that makes b, thus giving a + c (49) as the polynomial equation of our graph in terms of the -coordinate system As an eercise (eercise 4, following the net theorem), you can show that this choice is possible, and that by this choice ac AC 4 B
Chapter & Page: 4 8 Homogeneous Constant Matri Systems, Part II This and lemma 4 then yields: Lemma 4 Let A, B and C be real numbers, and assume the polynomial equation has a nonempty graph Then: A + By + Cy If AC 4 B >, the graph is an ellipse centered at the origin If AC 4 B <, the graph is a pair of hyperbolas? Eercise 4 a: Find the formulas for A, B and C when using the change of variables given by equation (47) to convert equation (46) to equation (48) b: Show that equation (46) reduces to equation (49) under the change of variables given by equation (47) if ( ) θ Arctan B if A C A C if A C c: Assume that θ is as above, and verify that AC 4 B ac π 4 Verifying Lemma 4 Now consider (t) v cos(ωt) + w sin(ωt) assuming ω is some nonzero real number, and v w v and w v w is a linearly independent pair of vectors whose components are real numbers Rewriting the above formula for in component form, we see that (t) v w v cos(ωt) + w sin(ωt) cos(ωt) + sin(ωt) y(t) v cos(ωt) + w sin(ωt) v w That is, (, y) is the point on the trajectory given by (t) if and only if cos(ωt) v w M where M y sin(ωt) v w (4) Since the two columns of M are the column vectors v and w, and {v, w} is linearly independent, we know (from linear algebra) that δ det(m) v w w v
Additional Eercises Chapter & Page: 4 9 This tells us M is invertible The inverse of M is easily found, and using it, we can then invert relation (4), obtaining cos(ωt) w w w w y sin(ωt) δ v v y δ v + v y From this and our favorite trigonometric identity, we have cos (ωt) + sin (ωt) δ (w w y) + δ ( v + v y), which, after a little algebra, simplifies to with A + By + Cy A (w ) + (v ) δ, B w w + v v δ and C (w ) + (v ) δ From theorem 4 we know this trajectory is an ellipse centered at the origin if AC B / 4 > Remarkably, when you carry out the details of computation, you get AC 4 B δ 4 [( w + v )( w + v ) (w w + v v ) > So, yes, the trajectory is an ellipse centered at the origin, and lemma 4 is confirmed? Eercise 4: Fill in the details of the computations in the above proof Additional Eercises 4 Find a general solution, in terms of real-valued functions only, for each of the following systems: 4 a b 4 y 8 y 4 c d 5 y y e f y 5 y z z 4 g y h y 8 z 4 z z z
Chapter & Page: 4 Homogeneous Constant Matri Systems, Part II i 4 5 8 6 4 4 j z y z 44 For each of the following systems: i Determine whether the critical point is a center or a spiral point, and describe its stability ii Sketch (by hand) a phase portrait a y 4 b [ 5 y c y 6 d 5 y e g 6 y 8 y f h 5 y 5 6 y 45 Let A a b c d where a, b, c and d are fied real numbers Show that the eigenvalues of A are purely imaginary (and, hence, the system A has elliptic trajectories) if and only if d a and a + bc < Further show that, if the above holds, then the eigenvalues of A are ±i bc + a 46 Let (t) be a nonequilibrium solution to a constant matri system A having elliptic trajectories, and consider the dot product Observe that the vectors and are perpendicular (and, hence, ) if and only if is pointing along one the aes of the elliptic trajectory of (t) (see figure 46) Using this observation with the fact that A : a Show that the two aes for the system s elliptic trajectories are the coordinate aes if the diagonal elements of A are zero b Find the slopes of the straight lines y m containing the aes of the elliptic trajectories for each of the following systems: 5 i ii 5 y y 5 iii iv y y
Additional Eercises Chapter & Page: 4 Figure 46: The and vectors for various points on an elliptic trajectory 47 Find a general solution for each of the following systems: a b y c d 8 y 6 e f 6 y g 5 h 4 8 y 4 8 y 6 4 9 6 y 4 y 4 4 y 48 For each of the following systems: i Describe the stability of the equilibrium solution ii Sketch (by hand) a phase portrait a y b y 4 c e g y y 5 9 y d f h y 8 9 y y 49 Find a general solution for each of the following systems: 4 a y b 8 z z z y z
Chapter & Page: 4 Homogeneous Constant Matri Systems, Part II c e g z z z y d z 4 5 y f z y h z z z z 5 5 y 5 5 z y z 4 5 y z 4 The main goal of this eercise is to verify lemma 44 on page 4 4 a Assume A is any N N matri, and let B A γ I for some scalar γ Show that (r, u) is an eigenpair for A if and only if (ρ, u) is an eigenpair for B with r ρ + γ b For the rest of this eercise, assume A is a constant matri, and set B A ri where r is an eigenvalue for A with corresponding eigenvector u [u, u T i Using the first part of this eercise set, verify that Bu ii Then, using the fact that Bu, find formulas for the components of B iii Using the formulas just found for the components of B, show that there is a vector a [a, a T such that Bv (u v u v )a for each v [v,v T iv Assume that the a just found is nonzero, and, using the above, verify that it is an eigenvector for both B and A, and determine the eigenvalue r for A corresponding to eigenvector a v Let (r, a) be as just found, and suppose {u, a} is linearly independent Show that r r vi Let (r, a) be as just found, and suppose a Show that there is a linearly independent pair {u, v} where (r, v) is an eigenpair for A c Let A, (r, u) and a be as above, and assume, in addition, that eigenvalue r has algebraic multiplicity two but geometric multiplicity one Using the above: i Show that a must be a nonzero constant multiple of u ii Verify that there is a nonzero w satisfying (A ri)w u with w not being a constant multiple of u
Additional Eercises Chapter & Page: 4 4 Assume r is the only eigenvalue for some real constant matri A Further assume that every eigenvector for A is a multiple of u, and recall from eercise 9 on page 9 9 that every point on the straight line parallel to u through the origin is a critical point for A a Describe the general solution to A b Describe the trajectories of the nonequilibrium solutions c Sketch a phase portrait for each of the following systems i ii y 4 y 9 iii iv y y 4 Let (r, u) be an eigenpair for an N N matri A Assume r has geometric multiplicity one but algebraic multiplicity three or greater We know that two solutions to A are given by where w is any vector satisfying a Set (t) ue rt and (t) [ tu + w e rt [A riw u (t) [ t u + αtw + w e rt and derive the fact that is a solution to A if α and [A ri w w b Assume the algebraic multiplicity of r is at least four, and derive the conditions for α, β and w so that is a solution to A 4 (t) [ t u + αt w + βtw + w e rt
Chapter & Page: 4 4 Homogeneous Constant Matri Systems, Part II Some Answers to Some of the Eercises WARNING! Most of the following answers were prepared hastily and late at night They have not been properly proofread! Errors are likely! a c sin(4t) cos(4t) + c cos(4t) sin(4t) b c cos(4t) sin(4t) + c sin(4t) cos(4t) cos(4t) sin(4t) c c cos(4t)+ sin(4t) + c cos(4t)+sin(4t) d c sin(t) cos(t) e t + c cos(t) sin(t) e t cos(t) e c sin(t) cos(t) e t sin(t) + c cos(t) sin(t) e t [ cos(t) [ f c sin(t) e t sin(t) [ + c cos(t) e t + c e t [ g c e 4t cos(4t) sin(4t) + c cos(4t) sin(4t) + c cos(4t)+ sin(4t) 4 cos(4t)+4 sin(4t) [ [ cos(t) h c e t + c sin(t) [ 5 i c 5 4 8 [ [ j c e t + c cos(t) [ sin(t) e t + c cos(t) 4 cos(4t)+4 sin(4t) sin(t) [ 5 [ sin(8t) e t + c 548 e t 4 cos(8t) + c sin(8t) cos(8t) cos( t/) cos( t/) sin( t/) cos( t/)+ sin( t/) e t + c 4 [ cos(8t) 4 sin(8t) cos(8t) e t/ + c [ sin(8t) sin( t/) sin( t/)+ cos( t/) sin( t/) cos( t/) e t/ 4a A stable center 4b A stable center 4c A stable center
Additional Eercises Chapter & Page: 4 5 4d A stable center 4e An unstable spiral point 4f An asymptotically stable spiral point 4g An asymptotically stable spiral point 4h An unstable spiral point 6b i m ± 6b ii m, 6b iii m, 6b iv m ( ± ( [ 5)/ 7a c [ + t ) c e t ( [ 7b c [ + c t ) t e t
Chapter & Page: 4 6 Homogeneous Constant Matri Systems, Part II ( [ 7c c [ + t ) c t+ e 9t [ 7d c [ + t+ ( [ c t+ 7e c [ + 6t+ ) c 6t e 4t ( [ 7f c [ + t+ ) c t e t ( [ 7g c [ + 5t+ ) c t e t/ ( [ 7h c [ + c t+ ) e t t 8a Asymtotically stable 8b Unstable 8c Unstable 8d Unstable 8e Asymptotically stable
Additional Eercises Chapter & Page: 4 7 8f Unstable 8g Asymptotically stable 8h Unstable [ 4 8 [ 9a c e 4t [ t ) + (c + c e 4t [ [ t [ t ) 9b (c + c + c 6t+ e t [ [ 6t [ 4t ) 9c (c + c + c t e t [ 4t [ [ t+ 9d c + c t + c 5t [ t [ [ 4t ) 9e (c + c + c 8t e t [ t 4t [ t [ ) 9f (c + c + c e t ( [ [ t [ ) 9g c + c t + c e t [ t [ t [ ) 9h (c + c t+ + c e t t b iv r r + u a u a a (t) c u + c [tu + w where w is any solution to Aw u b They are the straight lines parallel to u that do not pass through the origin
Chapter & Page: 4 8 Homogeneous Constant Matri Systems, Part II [ c i c [ + c t+ t c ii c iii c iv