Oubo Ma.t.emática Educacional 

\fol., -'· f!lí'2, Junio 2002 

JORDAN NORMAL FORM VlA ODE'S 

D e r e k H acon 
Departamento de Matemática P UC-Rio1 

R un Marques de Sao Vicente 225 1 
Rio de J aneiro, CEP 22453-900, 

Brasil. 
E-mail address: d erek@mat.puc- rio.br 

Abstra t : I n Lext books on differential equations the systcm of ordinary 
d iffcrential equatio ns with cons ta nt coeffic.ients X ' = AX is often solved by 
rcduction (by a n invcrt.iblc cha ngc o f variables .:r = PY ) to t.he s impler sys-
tem },., = JY wherc J is t hc J ord a n canonical form o f A . Here we do things 
t.hc Ol her way a round tmd deduce t.h e existence o í J and P by compa ri ng t;wo 
type; o í solut.ions of the system X' = AX . The proof provid es a straight.-
forwo.rd algorithm for ca.Iculating t.he matrices J and P above. Apar t. from 
~orne elem ntnry consid0rations on (fo rmal) solutions of systems of ODE's 
with const.rull. coeff1 cients, t.he ma in ingredient of lhe proof (nnd of t. hc re-
suhing algorit.hm) is onc wh ich comes up in othe r ap proaches, namely t. he 
rcduclion of polynomia l matrices (in one variable) LO dia.gona l form by row 
o..nd oolumn opcro.tions. 

l. Introductio n 
A wcll know 11 upp licat. iou o f J ordan normal form is to solving t.hc 

syst. m X ' = AX . Th is sys t.e m of ODE1 can also be solv l dircct.ly ussing 
1,hc class1t'al m thod of reclt 1ct.ion t.o diagonal form vin row and <:0 11111111 
op ntions and t.he pnrposc of t.his not.e is to s how how, conversely, t his 
soll 1t.ion lead , in a na t ura l way, to t.h e:<lst.enre of th J ordan nor mal form 
fo r t.he matrix A. 

Th idea is Lh following: the derivnLive DX o f a ny sol11 t ion X o f a 
syst.em (wi1 h rons1.nnt coeffci · nts) is n.lso a sol11Lion. A diagonal system has 
a wcU knowu bu.si. reln l.ive l.o w h ich Lh c mnt.rix of D is a J a rcian matri x J. 
R 111ruo11 o f DX = AX to diagona l form takes Lhis basis into a basis for 



60 Jordan N ormal Form via ODE's 

D X =AX rela t ive to which the matrix of D is also J . But DX = AX has 
a well known basis relative to wh.ioh the matr ix of D is A . Thus A and J 
a re s imi lar . 

We begin by considering a nilpotent rnatrix A. ln t his case one need 
onJy cons id er polynomi al solutions X(t). In t he general (not necessarily 
ni lpotent ) case, formal power series must be u.sed instead. But quest ions 
of convergence and differentiaibili ty oE solutions never arise. Thus 1 despite 
t he appearence of conce¡Dts relarnefil to differential equa tions , only elemen-
tary algebra.ic proper t ies (na:mely liHe&r ity and t he Leibniz rule ) of for mal 
differentiation D are a sed. 

2. Polinomial Solutions of Systems of ODE 's 
The d er ivat ive Di/> of tloe pol:ynomial tf>(t) is defin ed formally (w it hout 

a ny reference to limi ts ) fuy tille NS1!1 a l form ula . Jf F (x ) is a poly nomi al 
ma trix t hen , setting x equal to D, we get a ma trix operator F(D) a li of 
whose en tri es a re poly nom ials f(D ) in D . For example, if I is the ident it.y 
matri x t hen xi corresponds t o tloe cli fferen ti ation opera to r (also d enot ed by 
D) which has D 1s clown the d iagolll al a nd zeros elsewhere. Clearl y, for a ny 
F(x) , one has 

F(D)D = DF(D ). 

Therefo re, if F ( D )X = O t h ern F(D)DX = O as we!J. 
A basis (alias Íltnd amental m11trix ) for F(D) is a poly nomiaJ ma t rix <l> (t ) 

s uch t ha t a ny polynomi al solu t.io n X (t ) of F ( D )X = O is a unique linear 
combina Li o n (wi t h r ea l coeffi oien ts) of t he columns of <I> (in other words 1 
t. he columns of <I> a re a basis fo r t.he space of solu t io ns X of t he syst.em 
F ( D)X =O). Since each column of D<l> is t he d eri vati ve of a olu t ion it is a 
li near combina tion of t. he columns of <P . T hu s t here is a (nnique) constant. 
ma t rix M (say) s uch Lhat. 

D <l> = <l> M. 

In other words 1 A/ is t he mat.rix of D relative to t he basis <I> . Simi la rly, if el> 
a nd .V are two bases for F ( D) t hen t. her is a constant invertib le mat rix P 
s uch that. 

l!J = <l>P. 

Applying D, we have 

D -11 = D(<I>P) = D(<l>) P = <l> M P = '1/ P- 1/lf P, 



Dcrck Hacon 6 1 

80 thnt l hc mnt.rires oí D rclativ lo any t.wo bases far F(D) are simila r. 
Thl'r ar two cas wh re bases are w 11 known. First, if A is any 11 il pol c nt 
·onstant mntr i x th n 

1 
6 = l +tA+2t'A'+ .. 

is a polynomial ma l rix whi h is asily n to be o basi for D - A. lcarly 

oe = eA. 

ndly, if a d iagona l rn ntrix 'D(D) has a basi then nonc of its d iagonal 
C'ntnes can b zcro, fo r o t.h rwi t.h ·re would b infi ni tcly mauy !i n arly 
in lrp<'nd nt olut ions. onvcrscly, 1 t / (D) be a diagona l ent ry f 'D(D) of 
1 hr fonn Dlr+ high r pow r f D wh r k > O. Th n t.hc row vect.or 

(1 t ... (k¡:li )I) 
'"n ""'"' for f(D). 11ch bases mny be as.cmbled 11110 R basis 6. for 'D(D). 
Clt·arly 

D = D. J 

wlwn• J 1s d 1rN't s11111 of Jorclan blotk ·, arh ro1t.-.ist111g of 011cs jusi abovc 
111(' dt&Ronal and z r s I:; wh r . 

In li:l'lltrnl , R "'"is f r F ( D ) may ll<' obta1111~l by r l11ci ng F t,o dia-
gonaJ form by row nnd c' olu mn opcra1 ions. Th1 ml.'I hod, whid 1 is bRs 1 on 
long d1,'1MOn of poly noll\inl:;, is xplo.incd m varions tcx t.bo ks, for C'xtun¡ IC' 
[J orchm (\lol 3, S<'<'li n 14 1) a nd gocs as follows. 

l...t·1 g bí.' a nonzcro cnLry oí F . 13y • dmngmg rows oí P and t.hcn 
c·olumn: oí F W<.' 111ay u.ss11mr t hat g is in Lht· upprr 1 ft hand com er f F' 
(i t' m rh firs1 row nnd c·ol11 mn). lí g d1vicl~ rv<·ry nt.ry in ils row nncl 
<·olnmn th n , by row n11cl C' lumn perations. all tht• rntri~ in t.h first. row 
ami 1ht-" firs1 rolumn of F cxrcpt g may be r<.'dul''-'tl to zero. In olh r word 
P ma~· be r!'d11('('(l to 11 © G (say). therwi , F 1s red uccd lo n matri x wit,h 

nonzt--ro entry of 1 w N el grrc ami th pro<'Cdurr 1s l hen r p at.ed. Sine 
r hr dcgrre rannol be rocluecd bcyond í'C'ro, F will ev nt.11 nlly b rccl uccd 
to h JI (sny). rx t., // is a simila rly redurt'd and o on. In thc cnd , a 
diagonal matnx 'D is blni nC'd . Tlms therr urc poly norni o.I mul.rircs U(.c) 
nnd \ "(r) (w11h polynorn iL\I invrr ') s11rh 1ha1 

UPV= D 



62 Jordan Normal Form vio ODE's 

Since U and V are invertible, <I> is a basis for 'D(D) if and only if V(D)<I> is 
a basis for F(D). 

3. Jordan Normal Form for Nilpotent Matrices 
Let A be a nilpotent matrix. By reducing xi - A to diagonal form , one 

can calculate explicitly U, V and 'D such that 

U(D)(D - A)V(D) = 'D(D) 

Now D - A has a basis, aarnely e. So 'D(D) also has a basis. Hence ó., 
as defined above, is a basis for 'D(D). Thus V(D)ó. is a basis for D - A. 
Therefore 

V(D)ó. = 8P 

where Pisan invertible c0FJ.sta111t matrix. Applying D , we have 

8AP = D (8P) = DV (D) ó. =V (D) Dó. = (V(D)ó.) J = 8PJ 

Since e is a basis, this implies that AP = PJ, as required. 
lf needed , the matrix P may be easily calculated , because it is t he 

constant term of 8P and hence aJso of V (D)ó.. 

4. The General Case 
Here we switch from poly nomial solut.ions to formal power series so-

lut ions with D defined formally by the usual formul a. Many textbooks 
(for example J ordan 's Cours d'nnalyse) prove the basic result (due to Eu-
ler) [EulerJ that the polynomial /(D) has a basis consisling of f,tie"' for 
each factor (x - a)" of f and J¡t1 eªtsin bt and fit'eª'cosbt for each fac-
tor ( (x - a)'+ b2) ' where O ,,,; j < k . For diagona l 'D such bases may be 
asse mbled 111 lhe obvious way into a basis ó. for 'D( D ). Again , Dó. = ó.J 
where J is o Jordan matrix. The res!. of t.he argument. goes t hrough as in 
t.he n.ilpotent case.. 



Derek Hacon 63 

111 . Jordn.n, Cours d 'anaLysc, nu1 hicr. V11lrus 

l2J L . Eul r , De integmttone aequat1orum d1ffcren1talmm altionmi 
groduum, Opero Omnin Vol. XX II 1> 10 149