Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger han you may have expeced. This will become especially clear on Monday, when we sudy secion 5.7. Definiion Consider he linear sysem of differenial equaions for x : x A x where A n n is a consan marix as usual. If x, x 2,... x n is a basis for he soluion space o his sysem, hen he marix having hese soluions as columns, x x 2... x n is called a Fundamenal Marix (FM) o his sysem of differenial equaions. Noice ha his equivalen o saying ha X solves X A X X nonsingular i.e. inverible (jus look column by column). Noice ha a FM is jus he Wronskian marix for a soluion space basis. Example page 35 2: 5: general soluion Possible FM: general soluion: 4 2 3 x y 3 4 2 x y 2 3 2 5 6 2 3 2 3 6 x c e 2 c 3 e 2 e 2 v 3e 2 3e 2 3 v 2 c 2 e 5 2. 2e 5 e 5 2e 5 e 5 c c 2
Theorem: If is proof: Since x is a FM for he firs order sysem x A x hen he soluion o x A x IVP x x is a soluion o he homogeneous DE x x x x x x x is a linear combinaion of he columns of i A x. Is value a is x I x x. Exercise ) Coninuing wih he example on page, use he formula above o solve IVP x y x y 3 4 2 x y ans: x 3 7 e 2 3 2 7 e5 2
Remark: If is a Fundamenal Marix for x A x and if C is an inverible marix of he same size, hen C is also a FM. Check: Does X C saisfy X A X X nonsingular i.e. inverible? d d C C (universal produc rule... see las page of noes) A C A C. Also, X C is a produc of inverible marices, so is inverible as well. Thus X C is an FM. (Noice his argumen would no work if we had used C insead.) If is any FM for x A x hen X solves X A X. X I Noice ha here is only one marix soluion o his IVP, since he j h column x j is he (unique) soluion o Definiion The unique FM ha solves is called he marix exponenial, e A...because: x' A x x e j. X A X X I This generalizes he scalar case. In fac, noice ha if we wish o solve x A x IVP x x he soluion is in analogy wih Chaper. x e A x,
Exercise 2) Coninuing wih our example, for he DE x 4 2 y 3 wih A 3 4 2 and FM e 2 3e 2 x y 2e 5 e 5 compue e A. Check ha he soluion o he IVP in Exercise is indeed e A x. > > wih LinearAlgebra : 4 2 A 3 : MarixExponenial A ; #check work on previous page 7 e 2 6 7 e5 2 7 e5 2 7 e 2 3 7 e5 3 7 e 2 6 7 e 2 7 e5 (8)
Bu wai! Didn' you like how we derived Euler's formula using Taylor series? Here's an alernae way o hink abou e A : For A n n consider he marix series e A I A 2! A2 3! A3... Convergence: pick a large number M, so ha each enry of A saisfies a ij enry ij A 2 nm 2 enry ij A 3 n 2 M 3... enry ij A k n k M k k! Ak... M. Then so he marix series converges absoluly in each enry (dominaed by he Calc 2 series for he scalar e Mn ). Then define e A I A 2 3 2! A2 3! A3... k k! Ak... Noice ha for X e A defined by he power series above, and assuming he rue fac ha we may differeniae he series erm by erm, X A A I A A A 2 2 2 3 2 2! A2 A 3... kk A k... 3! k! 2! A3... 2 2! A2... A X. Also, X I. Thus, since here is only one marix funcion ha can saisfy X A X we deduce X I k k! Ak k k! Ak... Theorem The marix exponenial e A may be compued eiher of wo ways: e A e A I A 2 3 2! A2 3! A3... k k! Ak...
Exercise 3 Le A be a diagonal marix,... 2... : : :... n Use he Taylor series definiion and he FM definiion o verify wice ha e... e e 2... : : : : e n Hin: producs of diagonal marices are diagonal, and he diagonal enries muliply, so k k... k 2... : : :... k n
Example How o recompue e A for 4 2 A 3 using power series and Mah 227: The similariy marix made of eigenvecors of A 2 S 3 yields A S S : 4 2 2 3 3 2 2 2 6 5 3 5 so A S S. Thus A k S k S (elescoping produc), so e A 2 3 k I A 2! A2 3! A3... k! Ak... S I 7 2 3 2 2! 2 3 2... S e S k k! e 2 e 5 7 e 2 3e 5 k 2 3 2e 2 e 2 6e 5 2e 2 2e 5 7 3e 2 3e 5 6e 2 e 5 which agrees wih our original compuaion using he FM. e 5... S
Three imporan properies of marix exponenials: ) e I, where is he n n zero marix. (Why is his rue?) 2) If AB BA hen e A B e A e B e B e A (bu his ideniy is no generally rue when A and B don' commue). (See homework.) 3) e A e A. (Combine () and (2).) Using hese properies here is a "sraighforward" algorihm o compue e A even when A is no diagonalizable (and i doesn' require he use of chains sudied in secion 5.5). See Theorem 3 in secion 5.6 We'll sudy more deails on Monday, bu here's an example: Exercise 4) Le Find e A by wriing A D and using e D N where D A 2 2 2 2 2 2, N N e D e N. Hin: N 3 so he Taylor series for e N is very shor.
Universal produc rule for differeniaion: Recall he -variable produc rule for differeniaion for a funcion of a single variable, based on he limi definiion of derivaive. We'll jus repea ha discussion, bu his ime for any produc " " ha disribues over addiion, for scalar, vecor, or marix funcions. We also assume ha for he produc under consideraion, scalar muliples s behave according o s f g sf g f sg. We don' assume ha f g g f so mus be careful in ha regard. Here's how he proof goes: D f g lim We add and subrac a middle erm, o help subsequen algebra: lim f g f g. f g f g f g f g lim f g f g f g f g. We assume ha muliplicaion by disribues over addiion: lim f g g f f g. The sum rule for limis and rearranging he scalar facor le's us rearrange as follows: lim f g g lim Differeniable funcions are coninuous, so we ake limis and ge: D f g f g f g. f f g. The proof above applies o scalar funcion imes scalar funcion (Calc I) scalar funcion ime vecor funcion (Calc III) do produc or cross produc of wo funcions (Calc III) scalar funcion imes marix funcion (our class) marix funcion imes marix or vecor funcion (our class) This proof does no apply o composiion f g f g, because composiion does no generally disribue over addiion, and his is why we have he chain rule for aking derivaives of composie funcions.
Mah 228- Week Mar 27-29 (Exam 2 on Mar 3) Mon Mar 27: Use las Friday's noes o discuss marix exponenials Wed Mar 29: 5.6-5.7 Marix exponenials, linear sysems, and variaion of parameers for inhomogeneous sysems. Recall: For he firs order sysem x A x is a fundamenal marix (FM) if is n columns are a basis for he soluion space o he firs order sysem above (i.e. is he Wronskian marix for a basis o he soluion space). is an FM if and only if A and is inverible. e A is he unique marix soluion o X A X X I and may be compued eiher of wo ways: e A where is any oher FM, or via he infinie series e A 2 3 I A 2! A2 3! A3... k k! Ak... Example : We showed ha if hen e 2 2 2!... 2... : : :......... 2 2 2 2! 2 n...... : : :... n 2 2 2! n...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger han you may have expeced. This will become especially clear on Monday, when we sudy secion 5.7. Definiion Consider he linear sysem of differenial equaions for x : x A x where A n n is a consan marix as usual. If x, x 2,... x n is a basis for he soluion space o his sysem, hen he marix having hese soluions as columns, x x 2... x n is called a Fundamenal Marix (FM) o his sysem of differenial equaions. Noice ha his equivalen o saying ha X solves X A X X nonsingular i.e. inverible (jus look column by column). Noice ha a FM is jus he Wronskian marix for a soluion space basis. Example page 35 2: 5: general soluion Possible FM: general soluion: 4 2 3 x y 3 4 2 x y 2 3 2 5 6 2 3 2 3 6 x c e 2 c 3 e 2 e 2 v 3e 2 3e 2 3 v 2 c 2 e 5 2. 2e 5 e 5 2e 5 e 5 c c 2
Theorem: If is proof: Since x is a FM for he firs order sysem x A x hen he soluion o x A x IVP x x is a soluion o he homogeneous DE x x x x x x x is a linear combinaion of he columns of i A x. Is value a is x I x x. Exercise ) Coninuing wih he example on page, use he formula above o solve IVP x y x y 3 4 2 x y ans: x 3 7 e 2 3 2 7 e5 2
Remark: If is a Fundamenal Marix for x A x and if C is an inverible marix of he same size, hen C is also a FM. Check: Does X C saisfy X A X X nonsingular i.e. inverible? d d C C (universal produc rule... see las page of noes) A C A C. Also, X C is a produc of inverible marices, so is inverible as well. Thus X C is an FM. (Noice his argumen would no work if we had used C insead.) If is any FM for x A x hen X solves X A X. X I Noice ha here is only one marix soluion o his IVP, since he j h column x j is he (unique) soluion o Definiion The unique FM ha solves is called he marix exponenial, e A...because: x' A x x e j. X A X X I This generalizes he scalar case. In fac, noice ha if we wish o solve x A x IVP x x he soluion is in analogy wih Chaper. x e A x,
Exercise 2) Coninuing wih our example, for he DE x 4 2 y 3 wih A 3 4 2 and FM e 2 3e 2 x y 2e 5 e 5 compue e A. Check ha he soluion o he IVP in Exercise is indeed e A x. > > wih LinearAlgebra : 4 2 A 3 : MarixExponenial A ; #check work on previous page 7 e 2 6 7 e5 2 7 e5 2 7 e 2 3 7 e5 3 7 e 2 6 7 e 2 7 e5 (8)
Bu wai! Didn' you like how we derived Euler's formula using Taylor series? Here's an alernae way o hink abou e A : For A n n consider he marix series e A I A 2! A2 3! A3... Convergence: pick a large number M, so ha each enry of A saisfies a ij enry ij A 2 nm 2 enry ij A 3 n 2 M 3... enry ij A k n k M k k! Ak... M. Then so he marix series converges absoluly in each enry (dominaed by he Calc 2 series for he scalar e Mn ). Then define e A I A 2 3 2! A2 3! A3... k k! Ak... Noice ha for X e A defined by he power series above, and assuming he rue fac ha we may differeniae he series erm by erm, X A A I A A A 2 2 2 3 2 2! A2 A 3... kk A k... 3! k! 2! A3... 2 2! A2... A X. Also, X I. Thus, since here is only one marix funcion ha can saisfy X A X we deduce X I k k! Ak k k! Ak... Theorem The marix exponenial e A may be compued eiher of wo ways: e A e A I A 2 3 2! A2 3! A3... k k! Ak...
Example How o recompue e A for 4 2 A 3 using power series and Mah 227: The similariy marix made of eigenvecors of A 2 S 3 yields A S S : 4 2 2 3 3 2 2 2 6 5 3 5 so A S S. Thus A k S k S (elescoping produc), so e A 2 3 k I A 2! A2 3! A3... k! Ak... S I 7 2 3 2 2! 2 3 2... S e S k k! e 2 e 5 7 e 2 3e 5 k 2 3 2e 2 e 2 6e 5 2e 2 2e 5 7 3e 2 3e 5 6e 2 e 5 which agrees wih our original compuaion using he FM. e 5... S
Three imporan properies of marix exponenials: ) e I, where is he n n zero marix. (Why is his rue?) 2) If AB BA hen e A B e A e B e B e A (bu his ideniy is no generally rue when A and B don' commue). (See homework.) 3) e A e A. (Combine () and (2).) Using hese properies here is a "sraighforward" algorihm o compue e A even when A is no diagonalizable (and i doesn' require he use of chains sudied in secion 5.5). See Theorem 3 in secion 5.6 We'll sudy more deails on Monday, bu here's an example: Exercise 4) Le Find e A by wriing A D and using e D N where D A 2 2 2 2 2 2, N N e D e N. Hin: N 3 so he Taylor series for e N is very shor.
Universal produc rule for differeniaion: Recall he -variable produc rule for differeniaion for a funcion of a single variable, based on he limi definiion of derivaive. We'll jus repea ha discussion, bu his ime for any produc " " ha disribues over addiion, for scalar, vecor, or marix funcions. We also assume ha for he produc under consideraion, scalar muliples s behave according o s f g sf g f sg. We don' assume ha f g g f so mus be careful in ha regard. Here's how he proof goes: D f g lim We add and subrac a middle erm, o help subsequen algebra: lim f g f g. f g f g f g f g lim f g f g f g f g. We assume ha muliplicaion by disribues over addiion: lim f g g f f g. The sum rule for limis and rearranging he scalar facor le's us rearrange as follows: lim f g g lim Differeniable funcions are coninuous, so we ake limis and ge: D f g f g f g. f f g. The proof above applies o scalar funcion imes scalar funcion (Calc I) scalar funcion ime vecor funcion (Calc III) do produc or cross produc of wo funcions (Calc III) scalar funcion imes marix funcion (our class) marix funcion imes marix or vecor funcion (our class) This proof does no apply o composiion f g f g, because composiion does no generally disribue over addiion, and his is why we have he chain rule for aking derivaives of composie funcions.
Mah 228- Week Mar 27-29 (Exam 2 on Mar 3) Mon Mar 27: Use las Friday's noes o discuss marix exponenials Wed Mar 29: 5.6-5.7 Marix exponenials, linear sysems, and variaion of parameers for inhomogeneous sysems. Recall: For he firs order sysem x A x is a fundamenal marix (FM) if is n columns are a basis for he soluion space o he firs order sysem above (i.e. is he Wronskian marix for a basis o he soluion space). is an FM if and only if A and is inverible. e A is he unique marix soluion o X A X X I and may be compued eiher of wo ways: e A where is any oher FM, or via he infinie series e A 2 3 I A 2! A2 3! A3... k k! Ak... Example : We showed ha if hen e 2 2 2!... 2... : : :......... 2 2 2 2! 2 n...... : : :... n 2 2 2! n...
e... e 2... : : : : e n Example 2) If A is diagonalizable we showed ha we go he same answer for e A using A S S and he Taylor series mehod, as we did we did using he mehod. In fac, S e S, so his makes sense. and
How o compue e A when A is no diagonalizable. This mehod depends on he fundamenal fac abou how he generalized eigenspaces of a marix fi ogeher.... "Recall" (his is really linear algebra maerial, bu mos of you haven' seen i.) For A n n le he characerisic polynomial p de A I facor as p n k k k 2 m... 2 m Any eigenspace of A for which dim E k j is called defecive. If A has any defecive eigenspaces hen j i is no diagonalizable. (If none of he eigenspaces are defecive, hen my amalgamaing bases for each eigenspace one obains a basis for n (or n, in he case of complex eigendaa). However, he larger generalized eigenspaces G defined by do always have dimension k j. If bases for each G For each basis vecor v of G e j v A x e j v A j j E j G j nullspace j A j I k j are amalgamaed hey will form a basis for n or n.... one can consruc a basis soluion x o x A x as follows: e j I I A e j v A 2! j I v 2 x e A v e ji e A A 2! j I v 2 j I 2 2! j I v 2 j I 2 v A 2! A A k j j I v k j! j I 2 v j I 2... v j I 2 v... A k j k j! j I k j A v... j I k j v. Noes: The final sum is a finie sum because v nullspace A you've jus reconsruced j I k j! If v was an eigenvecor, x e j v Use he n independen soluions found his way o consruc a, and compue e A. Exercise Find e A for he marix in he sysem: x 3 x 2 x x 2
ech check: > wih LinearAlgebra : 3 A ; facor Deerminan A IdeniyMarix 2 ; A : 3 So he only eigenvalue is 2. > B A 2 IdeniyMarix 2 ; NullSpace B ; 2 2 B : () > bu E 2 is only one-dimensional. However, he generalized eigenspace G 2 nullspace A 2 I 2 will be wo dimensional: NullSpace B 2 ;, (2) (3) Use his generalized nullspace basis o consruc a basis of soluions o x A x and use he resuling o consruc he marix exponenial... > x e 2 IdeniyMarix 2 B. : x ; x2 e 2 IdeniyMarix 2 B 2 2 B 2. : x2 ; x x2 : ;. ; MarixExponenial A ; #hese las wo should be he same!! In fac, he las hree in his case e 2 e 2
e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 e 2 (4) > >
Variaion of parameers: This is wha fundamenal marices and marix exponenials are especially good for...hey le you solve non-homogeneous sysems wihou guessing. Consider he non-homogeneous firs order sysem x P x f * Le be an FM for he homogeneous sysem x P x. Since is inverible for all we may do a change of funcions for he non-homogeneous sysem: x u plug ino he non-homogeneous sysem (*): u u P u f. Since P he firs erms on each side cancel each oher and we are lef wih u f u which we can inegrae o find a u, hence an x u. Remark: This is where he (myserious a he ime) formula for variaion of parameers in n h order linear DE's came from... f "Recall" (February 24 noes):... Variaion of Parameers: The advanage of his mehod is ha is always provides a paricular soluion, even for non-homogeneous problems in which he righ-hand side doesn' fi ino a nice finie dimensional subspace preserved by L, and even if he linear operaor L is no consan-coefficien. The formula for he paricular soluions can be somewha messy o work wih, however, once you sar compuing. Here's he formula: Le y x, y 2 x,...y n x be a basis of soluions o he homogeneous DE L y y n p n x y n... p x y p x y. Then y p x u x y x u 2 x y 2 x... u n x y n x is a paricular soluion o L y f provided he coefficien funcions (aka "varying parameers") u x, u 2 x,...u n x have derivaives saisfying he Wronskian marix equaion y y 2... y n y y 2... y n............ y n y 2 n... y n n... u u 2 : u n : f
Bu if we conver he n h order DE ino a firs order sysem for x y, x 2 y ec. we have x y x x 2 y x 2 x 3 y x n x n y n x n y n p x y p x y 2... p n x y n f. And each basis soluion y for L y gives a soluion y, y, y,...y n T o he homogeneous sysem x x 2 x 3 : x n......... : : : : : p p p 2... p n x x 2 x 3 : x n : f. So he original Wronskian marix for he n h order linear homogeneous DE is a FM for he sysem above, so he formula we learned in Chaper 3 is a special case of he easier o undersand one for firs order sysems ha we jus derived, namely u f u f
Reurning o firs order sysems, if we wan o solve an IVP for a firs order sysem raher han find he complee general soluion, hen he following wo ways are appropriae: ) If you wan o solve he IVP x P x f x x The he soluion will be of he form x u (where u f as above). Thus so Thus x u u u u u x. s ds s f s ds. Then u u x x u u s f s ds. 2) If you wan o solve he special case IVP x A x f x x where A is a consan marix, you may derive a special case of he soluion formula above jus as we did in Chaper. This is sor of amazing! x A x f x A x f e A x A x e A f d d e A x e A f. Inegrae from o : e A x x Move he x over and muliply boh sides by e A : e s A f s ds x e A x e s A f s ds.