The Jordan Normal Form: A General Approach to Solving Homogeneous Linear Systems. Mike Raugh. March 20, 2005

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1 The Jorda Normal Form: A Geeral Approach to Solvig Homogeeous Liear Sstems Mike Raugh March 2, 25 What are we doig here? I this ote, we describe the Jorda ormal form of a matrix ad show how it ma be used to solve a geeral homogeeous liear sstem of ordiar differetial equatios x = Ax (.) where A= A is a square -b- real matrix with costat etries. We do t show how to derive the Jorda ormal form, sice that is a more advaced topic tha we are prepared to deal with here. You are alread familiar with the simplest Jorda form uder a differet ame. It is the diagoalized represetatio for a matrix with a full set of liearl idepedet eigevectors, a topic discussed at legth i Math 63. The previous Note etitled Geeralized Eigevectors itroduced the ext simplest Jorda form, amel the Jorda block revisited below. Recall that two approaches ca be used to solve Eq. (.) i case A is diagoalizable. A is diagoalizable iff A has a complete set of liearl idepedet eigevectors, sa, { v,, v }. The first approach is to form the matrix P = [ ] v v cotaiig the eigevectors as colums, the i Eq. (.) substitute x = P a equatio that determies because of the ivertibilit of P. As we shall see below, this results i a easil solved decoupled sstem of separate first-order liear equatios, leadig immediatel to the geeral solutio of Eq. (.). The secod approach begis with a trial solutio of the form x= e v, where the costat scalar ad costat vector v are chose to satisf the costrait imposed b Eq. (.), amel, x = e v= Ae v= e Av (.2) Sice, the secod ad last members of the precedig equatio impl that Av = v, we fid that Eq. (.) has a solutio x= e v for each eigevalue-eigevector pair (, v ). It follows that, i the assumed case i which we have a complete set of liearl idepedet eigevectors { v,, v } with correspodig eigevalues {,, }, the geeral solutio of Eq. (.) is give b the liear combiatio mrr/hmc/math 64 of 5 3/2/25; 5:3 PM

2 x= ce v + + ce t v (.3) t for arbitrar scalars { c c },,. Jorda ormal form. Not all matrices are diagoalizable. But, ext best, ever square matrix is similar to a matrix i Jorda ormal form. I other word, if A is a square matrix, the there exists a o-sigular square matrix S such that S AS J = (.4) where J is a square matrix i the ormal form, to be described. Moreover, if the matrix ad all its eigevalues are real, there is a real matrix S that ields the Jorda form. A matrix J i the Jorda ormal form is oe that is composed of blocks o the diagoal, with each block a Jorda block: J k ( ) = (.5) k k The i each block of J must be a eigevalue of the matrix A (hece it ma equal zero), the same eigevalue ma occur i more tha oe block, but each eigevalue of A must be preset i at least oe of the blocks. The diagoal of each block sits o the diagoal of the matrix J, ad J cosists of ol zeroes outside the Jorda blocks. If all the blocks are of size oe (i.e., for each block k = ), the A is diagoalizable. The rak J is k iff, otherwise the rak is k. of k Jk Notice that whe Jk is cosidered b itself, the geometric multiplicit of i is, ad the algebraic multiplicit of is k. These multiplicities are summed i the matrix A i the sese that, if A has a total of m blocks with a idetical eigevalue, the geometric multiplicit of i A is m, ad the algebraic multiplicit is the sum of the values of k for those m blocks. The geometric multiplicit of is the dimesio of the eigespace of, ad the algebraic multiplicit of is the multiplicit of as a root of the characteristic equatio for A. If k > for at least oe Jorda block i A, the matrix A is said to be deficiet the A does ot have a full set of eigevectors. Otherwise, if the algebraic ad geometric multiplicities of A are equal, i.e., all the Jorda blocks are of size oe, the the matrix A ad all matrices similar to A have a full set of eigevectors ad the matrices are all diagoalizable; such matrices are said to be o-deficiet. To repeat, a matrix A is diagoalizable iff it is o-deficiet, i.e., iff each Jorda block of A is odeficiet. mrr/hmc/math 64 2 of 5 3/2/25; 5:3 PM

3 Diagoalizable Matrices. So, ow, what does this have to do with differetial equatios? Let s start with the simplest case. Suppose all the Jorda blocks of A are of size k =. This is the case metioed above i which A is diagoalizable. I that case J i Eq. (.4) is a diagoal matrix cotaiig the eigevalues of A o the diagoal, ad S is a matrix of correspodig eigevectors. If we kow S ad J, there s a eas wa to solve the sstem Eq. (.), which we repeat here for coveiece: x = Ax (.6) We referred to this method i the opeig paragraphs above, where we used the letter P to deote the matrix of eigevectors. Just substitute x = S i Eq. (.6). Here s what ou get: S = AS = J (.7) But J is a diagoal matrix cotaiig the eigevalues of A. Writig this out, we get = = (.8) So the chage of variables from x to has produced a decoupled sstem. I other words, we ca rewrite the equatio as = (.9) = Each of these equatios (.9) ca be solved separatel, ieldig all possible solutios, ad ca be represeted i the matrix formulatio ce = = (.) c e,, are arbitrar costats. But x = S, so we ca get the geeral solutio of ODE (.6) b multiplig Eq. (.) b S: where the set { c c } ce x = S (.) ce mrr/hmc/math 64 3 of 5 3/2/25; 5:3 PM

4 The costats { c c },, ca be chose uiquel to solve a give iitial value problem, a fact made evidet b the fact that S is ivertible. Thus, i oe swoop, our derivatio of Eq. (.) gave us the geeral solutio of ODE (.6). Notice that this method ields the same solutio as give b Eq. (.3), but the method show here exploits the uderlig structure of A more explicitl. No-diagoalizable matrices. Now we re read to solve the sstem of differetial equatios for the case of a geeral matrix A. Agai, we suppose we have alread obtaied S ad J for the Jorda from i Eq. (.4); obtaiig S is the hard part, but we assume that it has bee doe to demostrate the utilit of the Jorda form. It is sufficiet to demostrate a derivatio for the case i which J cosists of a sigle Jorda block: J = (.2) Agai, let x = S, so that the trasformed Eq. (.7) looks like: 2 + = = (.3) + This sstem of equatios is ot decoupled. Your text calls equatios of this tpe a cascade. Usig e as a itegratig factor, the sstem is easil solved progressivel i reverse order, ieldig: = c e = c + c t e t = c+ c2t+ + c e (! (.4) mrr/hmc/math 64 4 of 5 3/2/25; 5:3 PM

5 where c,, c are arbitrar costats of itegratio. These solutios are similar to the fuctios used i the Note etitled Geeralized Eigevectors to itroduce the Jorda form i the first place! Assemblig these results i matrix form, we get c c c c 2 c2 c3 c t 2 = = e c c t c (! (.5) Usig x = S agai, we fid a explicit geeral solutio for Eq. (.6) as c c c c 2 c2 c3 c t x = S e c c t c (! (.6) It is apparet from the precedig represetatio that the solutio space of Eq. (.6) i this case is agai a vector space of dimesio. The techique for computig geeralized eigevectors discussed i our lecture o geeralized eigevectors ad i our text is hit-ad-miss. A reliable ad sstematic solutio, as we saw above, reduces to the problem of computig a Jorda form for the give matrix A. Techiques for doig the latter are the subject of advaced liear algebra or umerical aalsis. mrr/hmc/math 64 5 of 5 3/2/25; 5:3 PM