ODEs II, Supplement to Lectures 6 & 7: The Jordan Normal Form: Solving Autonomous, Homogeneous Linear Systems. April 2, 2003

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1 ODEs II, Suppleme o Lecures 6 & 7: The Jorda Normal Form: Solvig Auoomous, Homogeeous Liear Sysems April 2, 23 I his oe, we describe he Jorda ormal form of a marix ad use i o solve a geeral homogeeous liear sysem of ordiary differeial equaios wih cosa coefficie marix A. By he way, you are already familiar wih he simples Jorda form. I s he diagoal represeaio for a marix wih a full se of liearly idepede eigevecors, a opic discussed a legh i Mah 63. (Recall ha a marix is diagoalizable iff i has a full se of liearly idepede eigevecors.) Jorda Normal Form. No all marices are diagoalizable. Bu, ex bes, every square marix is similar o a marix i Jorda ormal form. I oher word, if A is a square marix, he here exiss a o-sigular square marix S such ha S AS J = (.) where J is a square marix i he ormal form, o be described. Moreover, if he marix ad all is eigevalues are real, here is a real marix S ha yields he Jorda form. A marix J i he Jorda ormal form is oe ha is composed of blocks o he diagoal, wih each block a Jorda block: J k ( λ ) λ λ = λ λ k k (.2) The λ i each block of J mus be a eigevalue of he marix A (hece i may equal zero or be a complex umber), he same eigevalue may occur i more ha oe block, bu each eigevalue of A mus be prese i a leas oe of he blocks. The diagoal of each block sis o he diagoal of he marix J, ad J cosiss of oly zeroes ouside he Jorda blocks. If all he blocks are of size oe (i.e., for each block k = ), he A is diagoalizable. To lik up our vocabulary from Mah 63 wih his represeaio of a marix i J λ is cosidered by iself, he geomeric mulipliciy of Jorda form, oice ha whe k ( ) λ i J ( ) k λ is, ad he algebraic mulipliciy of λ is k. These mulipliciies are mrr/hmc/mah 64 of 5 4/2/24; 4:42 PM

2 summed i he marix A i he sese ha, if A has a oal of m blocks wih a ideical eigevalue λ, he geomeric mulipliciy of λ i A is m, ad he algebraic mulipliciy is he sum of he values of k for hose m blocks. The geomeric mulipliciy of λ is he dimesio of he eigespace of λ, ad he algebraic mulipliciy of λ is he mulipliciy of λ as a roo of he characerisic equaio for A. If k > for a leas oe Jorda block i A, he marix A is said o be deficie - he A does o have a full se of eigevecors. Oherwise, if he algebraic ad geomeric mulipliciies of A are equal, i.e., all he Jorda blocks are of size oe, he he marix A ad all marices similar o A have a full se of eigevecors ad he marices are all diagoalizable; such marices are said o be odeficie. To repea, a marix A is diagoalizable iff i is o deficie. Diagoalizable Marices. Wha does his have o do wih differeial equaios? Le s sar wih he simples case. Suppose all he Jorda blocks of A are of size k =. The, as meioed above, A is diagoalizable. I ha case J i Eq. (.) is a diagoal marix coaiig he eigevalues of A o he diagoal, ad S is a marix of correspodig eigevecors. As we kow, some of he eigevalues may be repeaed. If we kow S ad J, here s a easy way o solve he auoomous sysem of equaios x = Ax (.3) Jus subsiue x = Sy i Eq. (.3)! Here s wha you ge: Sy = ASy y = Jy (.4) Assuig J is a diagoal marix coaiig he eigevalues of A, we ge y λ y λy = = (.5) y λ y λ y So Eq. (.3) has bee reduced o Eq. (.5), a decoupled sysem. I oher words, we ca rewrie he he laer as y = λy (.6) y = λ y Each of hese equaios (.6) ca be solved separaely, yieldig all possible soluios, ad mrr/hmc/mah 64 2 of 5 4/2/24; 4:42 PM

3 y ce y = = (.7) y c e,, are arbirary cosas. Bu x = Sy, so we ca ge he complee soluio of ODE (.3) by muliplyig Eq. (.7) by S: where he se { c c } The cosas { c c } ce x = S (.8) ce,, ca be chose uiquely o solve ay give iiial value problem, a fac made evide by he iverabiliy of S. Thus, i oe swoop, our derivaio of Eq. (.8) gave us he complee soluio of ODE (.3). Ad, sice S is o-sigular, he cosas are arbirary, ad oe of he expoeials i Eq. (.8) ca vaish, i follows ha he soluio space is of dimesio. Thus we have proved he fudameal exisece ad uiqueess heorem for his sysem, ad have derived a explici soluio. Noice ha we did o come o his soluio hough a rial soluio; isead, we exploied he uderlyig srucure of A explicily. Nodiagoalizable Marices. Now we re ready for he case of a geeral marix A. Agai, we suppose we have already obaied S ad J for he Jorda from i Eq. (.). I is sufficie o demosrae a derivaio for he case i which J cosiss of a sigle Jorda block: λ J = (.9) λ λ Agai, le x = Sy, so ha he rasformed Eq. (.4) looks like: λ λy+ y2 y y = λ = (.) λ y y y + y λ λy We call Eqs. (.) a liear cascade. Usig e as a iegraig fac, he sysem is easily solved i reverse order, oe equaio a a ime: mrr/hmc/mah 64 3 of 5 4/2/24; 4:42 PM

4 y = c e ( ) y = c + c e y = c+ c2+ + c e (! ) (.) where { c c },, are arbirary cosas of iegraio. Re-assemblig hese resuls i marix form, we ge c c2 c c y c2 c3 c y 2 y = = e (.2) c c y c (! ) Usig x = Sy agai, we fid a explici complee soluio for Eq. (.3) as c c c c 2 c2 c3 c x = S e c c c (! ) (.3) I is appare from he precedig represeaio ha he soluio space of Eq. (.3) i his case is agai a vecor space of dimesio. Ad, because S is iverible, he cosas ca be back-solved o saisfy uiquely ay give iiial codiios. Moreover, by amig he colums of S geeralized eigevecors, we ca see his of he moivaio for he rial soluio suggesed a he ed of he oes for Lecure 7: 2 3 x = δ+ γ+ β+ α e λ (.4) 2! 3! For example, supposig ha A= A 33 ad ha αβγ,, are colum vecors of S: Eq. (.3) yields S = [ γ, βα, ] (.5) I a geeralizaio of omeclaure, we have desigaed he oe rue eigevecor amog he colums of S (i.e., he h colum) a geeralized eigevecor. mrr/hmc/mah 64 4 of 5 4/2/24; 4:42 PM

5 2 x = cα+ c2( β+ α) + c3γ+ β+ α e λ (.6) 2 verificaio of which is a good exercise o es your udersaig. The precedig discussio proved already ha Eq. (.6) represes he complee soluio for he special 3 3 case of Eq. (.3) ha we are cosiderig. Bu we ca affirm α, β+ α, γ + β+ α is liearly idepede his i aoher way. The se of vecors { ( ) ( )} because, by hypohesis he se { αβγ,, }, i.e., cosisig of he colums of S, is liearly idepede. Therefore, he Wroskia of he hree fucios 2, ( ), eα e β+ α e γ+ β+ α (.7) 2 evaluaed a = cao vaish, provig ha he fucios are liearly idepede. The coefficies { c, c2, c 3} are arbirary, hece he expressio for x i Eq. (.6) is a geeral soluio for Eq. (.3) i his case. The echique for compuig geeralized eigevecors discussed i Lecure 6 ad i your ex is hi-ad-miss for algebraic mulipliciy greaer ha wo. A reliable ad sysemaic soluio, as we saw above, reduces o he problem of compuig a Jorda form for he give marix A. Techiques for doig he laer are he subjec of advaced liear algebra or umerical aalysis ad are beyod he scope of Mah 63 ad 64. mrr/hmc/mah 64 5 of 5 4/2/24; 4:42 PM