Reductien of Systems of Linear Differential Equations to Jordau Normal Form.

by MAl~Y L. CARTWRIGHT (Cambridgo, Inghilterra)

To Giovanni Sansone on his 70 ~h birth day.

Summary. - Various me~hods are discussed of finding a non-singular matr ix P such that P A P - i ~-J , ~vhere J is the JORDAN normal form of A, ~vith special reference to the problem of reducing the system of equations x-~- Ax to the form y ~. Jy, where y ~-- Px.

1. Let A be an n X n matrix with real elements aii and ~c and y column vectors with coordinates (xl, x~ , . . .x , ) (y~, Y2, ... y , ) respectively. It is well known from the theory of matrices that there is a non-s ingular matr ix P with elements p~] such that

(1) P A P -1 ~ J

where J is the 3ORDAN normal form of A which we shall describe in detail present ly and p - 1 is the inverse of P. This result is used extensively in connection with the system of differential equat ions (1)

d=Ax,

where the dot denotes differentiat ion with respect to t and the elements of A are constants. For pu t t ing y ~ P x we have

(3) ~] - - PJ~ ----- P A x - - P A P - l y - - J y

and this form, or the corresponding real form (~) obtainable from it, enables solutions of (2) to be expressed in a form which is convenient for many purposes. In par t icular it is used for the discussion of stabil i ty of l inear

(i) See CODDINGTON awl LEVINSON, Theory of O:dinary Differential Equations (1955), Chapter 3, G. SANS()NE and R. CONTI, Equazioni Differenziali Non Lineari (Idomo 1956), Chapter V I I I , ¥ . V. ~EMITSKY and V. V. STEPA~OF, Qualitative theory of differential equations, 2nd ed. Moscow 1949.

(e) See SASSoSE and CONTI, p. 5~3 footnoto l l .

148 M . L . OARTWRIC, IIT: Ned~w, tion of ,qy,~tems of Li~oar Dif]erc~tlal~ etc.

and nearly l inear systems, at~d systems with optim~d discontinuous forcing terms (a).

The usual methods of establishing (1) are by no means easy, and give little guide in finding the matrix 19 for any part icular system (2 )o r in reducing it to the form (l). The object of this paper is to discuss various special results and methods for f inding P, and also, by showing that some of the algebraic processes used in the proof of (1)cor respond exactly to simple manipulat ions of the system (2), to suggest a variety of ways of obtaining (3) some of which may give very quick results for fairly small values of n. In any ap})lication it has to be considered what is the essential information required and iu many cases this is obtained more easily from the classical e lementary methods of solving (2 )which are very much simpler than the process of finding J. It is hoped that this paper will help to link the two more closcl.y.

2. Jordan Normal Form. There are two important types of normal or canonical form, JORDAN normal form and another which we shall call standard normal form which we shall also use, and is in some respects more funda- mental. Matrices A and B are equivalent if there exists a non-s ingular matrix P such that P A P -~---B, and a set of canonical n X n matrices is such that (i) every n X n matr ix is equivalent to a canonical matrix, (ii) two distinct canonical matr ices are not equivalent.

The JORDAN normal form J is the one which is useftil in stabili ty problems, and depends part ly on the roots of the characteris t ic equation

~1) ~() . ) ---- H A - - ~.I [I = o.

Suppose that ).~, ).2, ... ks are the distinct roots of (1), and that )~ is repea. ted N~ times so that

s i2) "~ ., _N's ~ . ~.

('~) E. COD[nNC, TON and LEVINSON, Perturbations of linear systems with constaut coeffi- cients possessing periodic solutions, Contributions to the theory of non-l inear oscillatio~ts, Vol 2, ed S. L~FSCHETZ, (( Annals of Maths Studies>>, 29, 1952, 19-35, I-L I~. TUItmTIN, Asymptotic expansions of solutions of systems of ordinary differential equations containing a ~arameter, loc. cit, 81-116~ P. MENDELSON, On phase portraits of critic,1 points in -space, Contributions to the theory of non-l inear oscillations, Vo]. 4, ed S. LEFSOItETZ, ¢ Annals of Maihs Studies ~>, 41 0958), t67-199, D. BusrIAw, Differential equations wi th a discontinuous forcing term. Experimental Towing Tank, Stevens Insti tute of Technology ~o. 469 (i953), Z. SZ.~YDT, On th degree of regularity of surfaces formed by the asymptotic integrals of differential equations~ <( Annalos Polonici Mathematici >>~ I I 2 (1955), 294-313.

~. L. CARTWRIGt_IT: Rcductio~b of Sys tems of Linear Dif]ercntial, etc. 149

Corresponding to each root ~s repeated Ns times there are K(s) square blocks along the principal diagonal of J of the form

(3t

I k~, 1, 0 , . . .0 /

0, ~ , t, 0

d~, ~ ---- 0, 0, ),~, ... 0 ,

• . , • , , • ,

O, O, O, ... )~

and the remaining elements of J are zero. If n~(k) is the number of rows of J~, i~, the 1 ~ ns(k) ~ h~, and

K(s)

k - ~ - I

Hence we may write

J = i J,1

0

o I...o t ! ...o

0 ... Js, ~c(s) /

but changing the order of the blocks J~,~: does not change the JORDAN normal form. For it is easy to see by interchanging yi and yj that the i t~ and ]th row in the system 1 (3) can be interchanged, provided that the i th and ith column of J are interchanged at the same t ime; the operation is equivalent to putt ing ,7*----QJQ-~, where Q--: (q~h) interchanges the ita and j th rows and Q-1 interchanges the i tu and jth columns. This is easily verified by putt ing qq ~ q]~ = 1, qa ~ 1 for 1:4: i, 1:4: j, and q~h ~ 0 otherwise. Hence J* is equivalent to J, and we may regard J as known if we know K ( s ) a n d also us(k) for k ~ 1, 2, ... K(s ' , s ~ 1, 2,.. . S. It is not sufficient to know K(s) unless this is itself determines us(k).

3 . The ease of dis t inct roots. - I f S - ~ n so that none of the roots of 2(l) are repeated, N~ ~ - R ( s ) - : 1 for aI1 s, and so ~ ( l ) : = l. All the diagonal blocks consist of single elements and J is deter'mined at once. The matrix P consists of the n row vectors 1)8' such that

p~'(A -- ~ I ) = O.

150 M . L . CARTWRIGIIT: Reduction of Systems of L i , ear Differential, etc.

Since the ),~ are all d is t inct , it is easy to show that the p;, ' s~-- 1, 2.. . n.. are l inear ly i ndependen t and the re fo re

is non - s ingu l a r .

P ~ / Pi' 1 I I P2'

P . ' / \

4. '['he Rank of A - - ) . , 1 . - Suppose that the r ank of A - - ) , ~ I i~ r{s). Then 0 < r ( s ) ~ n - - 1 , and it is wel l known that there are exac t ly L(s)-~

t - - n - - r(s) l inea r ly i n d e p e n d e n t vec tors 1'~,1, t ----- 1, 2, ... L(s), can be found such that

(1) p ' ~ , t ( A - - X ~ I ) = O , s = l , 2 , . . .S .

Then, if the coord ina te Yi = Y~,z-- 'p'~,tx, by (1) we have

For each ,~ tile set of vec to r s P'~,t sat is t)ying (1) for 1-----1, 2,... L(s), are l inear ly independen t , and also the who le set is l inea r ly independen t . Fo r if not, there exis t n u m b e r s ~,,,~ not a, II 0 such that

S L(s) (3) v v , .. , Ft~,ll,~,~= O.

8=-I ~=I

Consider first the ease s = 1 ; by (1)

8 L(s) A = II ~ p .~ , l~ , '~ , l (A-- ; . t I )=O, s > 1,

S 1,(1) - - 1[ (Xl Xt) " '

1=-2 l 1

H e n c e by (3) S L<s) S

O ~ .,v ~ ' II (A, --)~t.I) ~s, zPs, z 8 -:I Z--I t~2

L(l) S

l=l t=2

3[. L. CARTWRI(~HT: Reduetio~ o] System.~, of Linear Dif]erential; etc. 151

Since ~ --#: ).t, t - - 2, 3 ... S, we have

L(~) J E ~.,z~ 1,~-- O,

l=:l

which con t rad ic t s the fact that 1/~,~, un les s t~ ,z - - 0 for l ----- 1, 2, ... L(1). we find that (2) is fa lse unless ~ , z - - 0

l - - 1, 2, ... L(1) a re l inear ly independent , Repea t ing the process for s--" 2, 3~ ... S, for all s and l which gives a cont radic t ion .

5. l ) e t e r m i n a t i o n o f J . - S ince there are exac t ly L(s) l inear ly i ndependen t vec tors P'~:z co r respond ing to each ~.~, s - - : l , 2 , . . . S , there are exac t ly L(s) equa t ions Ys,~ -~-- )-~Y~,~, w h e r e y,~,~--~ p's, zx . Since the last, and only the last, r ow of each b lock Js, k of the form 2 ( 3 ) c o r r e s p o n d s to an equa t ion of this type, and s ince the vec tors p~, k co r respond ing to these rows in P are l inear ly independen t , it fol lows that L ( s ) - - K ( s ) . ]n cer ta in cases the form of J can be d e d u c e d f rom this, Fo r ins tance , if

S (t) L - ~ ~ K ( s ) - - n ,

S::I

each b lock J~,k only has one row. For L is the total n u m b e r of b locks which is less than n if any b lock has more than one row. If K ( s ) - - 1 for all s, there is only one b lock co r r e spond ing to each s, and by 2 ( 4 ) n s ( 1 ) : A ~ . Again if K (s) - - _N~ - - 1 , it fol lows f rom 2(4) that n s ( k ) ~ 1 for all k excep t one for which n.~(k) ~ 2. In these cases the form of J is comple te ly de t e rmined by the roots ks, 8-----l~ 2 , . . . S of the charac te r i s t i c equa t ion 2(1) and their mul t ip l i c i ty hT~ and the r ank r(s) of A - )~,I for s ~--1, 2, ... S. S u m m i n g up we have the fo l lowing result .

TKE()nE~ 1 . - (i) I f r(s) is the r a n k of A - - k ~ I for s ' - ' l , 2 , . . . S , then r(s) - - n - - K(s) , (it) I f (1) holds, then the m a t r i x P which consists o f n l inearly independent row vectors ~/~, k, k - - 1, 2 .... K(s), s - - 1, 2, ... S sa t i s fy ing 4(1) sat is f ies 1(1) (iii) I f for each s - - 1, 2, ... S, either K(s) --~ 1 or K(s ) ~ iV~ - - 1, then J is completely de termined by the numbers ).~, K(s), s ---- 1~ 2 . . . . . I n par- t icular i f n ~ 3, lhen J is de termined by the numbersk~, K(s).

II' n ~ 3, the only poss ib le values of K(s) are 1, 2, 3 and so e i ther K ( s ) ~ 1~ or K(s ) ---- 2 or 3 in wh ich case K(s) ~ 7N~ - - 1 be c a use /V~ ~ 3.

6. The 6~eneral Case. - Suppose that for f ixed s the b locks J~,k are a r r anged in n o n - d e c r e a s i n g order of the n u m b e r of their rows, so that

(1) 1 < n~(1) ~ n~(2) ~ ... < n~(K(s)).

152 M . L . (~ART'WRIG]IIT: R e d u c t i o n of ,qysfem.~ of Li~war l)i f fereJtfial , etc.

For a f ixed s let K(s , m) denote the n u m b e r of blocks J:,k wi th at least m rows so that

I f(s) = K ( s , 1) ~ K(s , 2) ~ ... ~ K ( s , n ,K(s ) ) ~ 1,

and K(s , m ) = 0 for m ~ n~(K(s)). Then

n~(k,) = m, K(8) - - K(8 , m) + 1 ~ k ~ K( s ) - - I ~ (s, m + 1),

m ---- 1, 2, ... n~(K(s)).

Hence the form of J is comple te ly de te rmined by the roots ~ , s = 1, 2,. . . S of ~( ) . )= 0 and the number s K ( s , m), m = 1, 2,. . . n~(K(s)), s = 1. 2, ... S. For these de t e rmine the numbers n~(k), k = 1, 2, .. K(s), s - - i , 2, ... S. The numbers K ( s , m) can be de te rmined by f ind ing P as we shal l see in the nex t section.

7. The D e t e r m i n a t i o n o f P. - Le t l;~,~(m) denote the vector fo rming the row of P cor responding to the mth row of tile block Js,~. S ince 1(1) can he wr i t t en in the form P A =: J P , for each s we have

(1) p'~,~,(m)(A - - ),~I) = I/~,7~(m - - 1),

where 1"~.. ~(0) : O, m = 1, 2, ... n~(k), k - - 1, 2, .., K(s). Hence for m : 2, 3, ... n~(k), k = l , 2, ... K(s ) , we have

(2) I ~, ~(m) (A p'~, ~ (t),

where /;~,~(1) sat isf ies 4(1). The pa r t i cu la r equa t ion in 1(3) corresponding to (1) is obta ined by pur l ing y~,1,(m)=p'~,l,(m)x. Fo r then

Since the ma t r ix A - - ~ . s I is singular~ (2) is en ly sat isf ied if p'~,~(1) sat isfies cer ta in cons is tency condit ions. I t fel lows from the genera l theory of tile ,tORDA,~ normal form tha t vectors sa t i s fy ing (1) and (2) exist , and, a l though it is fa i r ly easy to ve l i fy this in cer ta in special cases, a d i rec t proof in the genera l case does not seem to be easy. In par t icu lar , if K ( s ) - ~ N , so tha t n~(k)----- 1 for k = 1, 2, ... N.~, equa t ion (2) is not needed. For the case K ( s ) - - 1 , m-~---2 Dr. J. A. TODD has shown me a s imple proof of the exis tence of p'~,~o(2) depend ing only oil the fact that ~ ¢ ( ~ , ) - - ' ( ( ~ ) . = 0 . In genera l ).~ is a root of the Character is t ic equa t ion 2(1) repea ted 2V~ t imes so that

(3 ) ~ ( ~ ) = ~ , ( ~ ) = . . . _ ~ N ~ - ~ ( ~ , ) - - O,

~-[. L. CART\VRIGIIT: Reduction of ,gystcms of Linear Differential, etc. 153

and it is in v i r tue of these re la t ions tha t the vector p'~,k(m), 2 ~ m ~n~(k) exist. The n u m b e r of equa t ions in (3) is N~ which by 2(4) is equal to the total n u m b e r of rows in the blocks Js, k for f ixed s and k ~ 1, 2,. . .K(s), and the equa t ions (3) also affect the r ank r(s) of A - - ) . J which gives the number K(s) = n - - r(s) of b locks J~, ~.

S u m m i n g np we have

THEOREM 2. - The matrix P for which 1(1) holds consists of linearly independent vectors p's,~(m), m-----1, 2, ...n,(k), k = 1, 2,.. .K(s), s---~ 1, 2 , . . .S , satisfying (1) , where 1/~,~:(0)--O.

Since the vectors p',,~(m) are all l inear ly independen t the set of K(s, m) vectors cor responding to the m rows of the K(s, m) blocks wi th at least m rows are l inear ly independent , and the set of vectors p'~,k(1) in (2) from which they are obta ined are l inear ly independent . S ince the la t te r sa t is fy 4 (1 )we have the fo l lowing resul t .

THEOREM 3. - l~br fixed s and m there are K(s, m) linearly independent vectors q'~,l~(m), K ( s ) - K(s, m ) + 1 ~ k ~ K(s), such that

(4~ q',. ~On)(A - - k f l ) ~-- O,

and i f m ~ 1 the equations

(5) t'~. k(t) (A - - )~I~ *-~ ----- q'~, k(m), l - - 2, 3, ... m

are also satisfied by exactly K(s, m) linearly independent vectors t 's ,k(l)for each l. I f for each m we number the vectors q's,l~(m) so that i f K ( s ) - - -- K(s, m) ~ 1 ~ k ~ K(s) - - K(s, m ~- 1), then (5) holds for 1 ---- 2, 3, ... m but not for m - ~ 1, the vectors p'~,~:(t), 1 - - 1 , 2 , . . .m, K ( s ) - - K ( s , m ) ~ l ~k~_~

K ( s , m - b l ) ob ta inedbypu thngp~ ,k (1 ) - - ' m ' l ... - - " ' q ~ k ( ) , P ~, ~( ) ----- ~'~, ~(l), 1 - - 2 , 3 , m ,

in (5) are those for the blocks J~, ~, with e~actly m rows.

The procedure of de t e rmin ing _P is therefore to f ind f irst the vectors q~,l~( ) sa t i s fy ing (4) for m 1. Since the r a n k of A )~sI is r(s), there are K(s) such vectors, and the vectors q's,k(2) are such tha t

K(s) ' 2 ~ 2 ' (61 q ~, k( ) Z ~,~(s, )q,,~(1),

and also (5) is sa t i s f ied for l - - 2 , m ~ 2, by a ce r t a in n u mb e r of l inea r ly independen t vectors t's,k(2) the n u m b e r being posit ive unless K ( s ) - - N , . The m a x i m u m n u m b e r of l inear ly independen t vectors t'~,k(2) which can be obta ined for f ixed s is K(s~ 2). W h e n this has been found we n u m b e r the vectors

Annali di Matematica 20

154 M . L . (~AB.TWR1GIlT: I~cduction of Sy.~tems of L inear Di]fcreutial , etc.

q'8, k(2) f rom k - - K(8) -- K ( s , 2) -{- 1 to K(s), and put

! 1 t ~*o P~,k( ) - - q s , k(1), k - " 1, 2, , K ( s ) - - K(8, 2),

w h e r e these vec tors q'8, k(1) a re any subset wh ich are l inear ly i n d e p e n d e n t of the vec tors q'~,k(2), K ( s ) - - K ( 8 , 2) -{ -1 ~_ k ~ _ K ( s ) . The process is then

' 2 ' 1 ' ' r epea t ed wi th the vectors qs, k(') in place of the vectors qs, k( ) and qs, k(3)in p lace of q'~.l.(2) wi th 1 - - 3 , m - - 3 in (5). Tile m a x i m u m n u m b e r of l inear ly i n d e p e n d e n t vectors q'8,~(3) obta inable is K(8, 3), and n u m b e r i n g them from k - - K( s ) - - K ( s , 3) --[- ] to K(s) , we put p's.l,(1) = q's,.~(2), K(s ) - - K(8 , 2) + + 1 ~ k ~ K ( s ) - - K ( s , 3), w h e r e these vec tors ,?'~,d2) a re any subset wh ich a re l inea r ly i n d e p e n d e n t of the vectors q'~.k(3). Conti~ming in this w a y we obtain all the row vectors of _P for each f ixed 8, and thus the whole matr ix .

I t should be r e m e m b e r e d that q's,t:(m) m a y be the same for m - - 1, 2,... ... n~(K(s)). This is in fact the case when K(s ) ---- 1,, and there is only one vector q'~,~.(1) and only block d,,~ to consider . In this case we have•

OO:ROLLAICY OF THEOI{E):[ 3. - If K (s) - - 1 for s - - i , 2, ... S, then K(s , m) - - 1 for m - - 1 , 2, ... n , ( K ( s ) ) - - N ~ , there is only one l inear ly i ndependen t vector q's, ~(1) s a t i s f y i n g (4), a n d p'~,~(1) --- q'8.1(1), p'~, ~(l) - - t'8,~(l), 1 : 2, 3, ... N s , where

t'~, ~(l) sat is f ies (5) w i t h m - i , form the rows o f t ).

8. A s ingle d i f f e ren t i a l equa t ion . - The d i f fe ren t ia l equa t ion

(1) a

( D" -f- bl D ' - ~ -t- ... + b,,) x~ = O, D - - ~l-t '

can be expressed as a sys tem of the form 1 (2 )by pu t t ing x ~ - - x ~ + l , i - - - - 1 , 2 . . . n - - 1 . We then have

(2) x - - B.r~, B - -

O, 1, O, , . . . , 0

O, O, O, , . . . , 0

O, O, O, , . . . , 0 "

• • * • • * .

- - b n , - - b ~ - l , - - b n - 2 , . . . , - - b l

This type of m a t r i x has very specia l p roper t ies in re la t ion to JORDAN normal form, and, as we shall soe in § 10 s t anda rd normal form is base on it. In fact f rom m a n y points of v iew it is more logical to deduce the JORDA~

~[. L. CARTY~rRIGIIT: Redu(~tion of Sys tems of Linear Di.fferential, etc. 155

normal form of a general system from that of matrices of the type of B than to treat the equation (1) as a special case of a general system after reducing the general system to J-ORDAN ,o rma l form as some authors do.

The character is t ic equation of B is

(3)

and the JORDAN normal form of B has only one block of N, rows of the type 2(3) corresponding to each distinct root ).~ of (3) repeated _N~ times. Hence J is completely determined by the roots of (3). This algebraic resnlt can be verified by the elementary theory of differential equations. For since

N, (D - - )~,)~V,(ea-2v,) = e)~tD~v,, where v, --- ~ C,.et ~-~,

/C=1

and since (1) is linear, we have

s 14) x~ = E e).~ ~ E G, kt k-~,

S~l k~l

where Cs, e, k - - 1 , 2 , . . . 1 , , s = 1, 2, . . .S, are constants. Now if J contains two blocks J~.~, J.~,2 corresponding to the root ~ of (3), there are two equa- tions Y~= X~Yi, YJ = ):,Yi in the system y = Jy . Then since y ~ = CieZ2, y ] - Cie~ , y~ and Yi are not l inearly independent, and so the inverse of the transformution y ~ P x such that P B P -~ ~- J gives x~ as a l inear combination of y~, y~, ...y~ containing at most n - - 1 arbi trary constants which by 2(2) contradicts (4).

The determinat ion of the matrix P for B is not part icular ly simple. The inverse P-~ can be wri t ten down (4) in terms of the roots )~,, s - - 1 , 2,... S, in it the element of the N ~ - - i th row and i th column of the N s rows, and colnmns corresponding to )~ is the coefficient of )~ is ()~ + ~,,)i-~. An al ternat ive method is given by the corollary of Theorem 3. In some cases it may be easier to obtain these equations by eliminating the constants Cs, k in (4) from ~ , and x~+~=x i , i----1, 2 , . . . n - - 1 .

9. Polynomial Matrices. (~) - The problem of reducing the matrix A to JOaDA~ normal form is often approached by applying the theory of polyno- mial matrices to A - ~,[ as a polynomial in ~. because ttie operations involved

(4) See J. A. TODD, Projective and analytical geometry, (London 19~:7), 166. (~) The two following sections are based on O. SCHREIER and E. SPERNER, Introduction

to Modern Algebra and Matrix Theory, translated by M DAVlS and ~. :[~AIJSNEr~, (2gew York 1951) 344-371 where further details of the algebraic processes will be found.

156 3[. L. CARTWRI(:II'r: Redueti¢m, of Systems of Lim'ar DiffcrcJttial, etc.

are simpler. For in 1(1) it is the matrix P on one side and its own inverse on the other which are used in the equivalence relation, and in 1(3) any operation involving rows also involves a corresponding change of columns and vice-versa. On the other hand the polynomial matrix A - ),[ is similar to any matr ix E(k) such that

(1) E0,) ---- P(),)(A - - ),I)Q(~),

where P(),) and Q0,) are any two matrices whose elements are polynomials in ), such that their determinant I[ P0,)II, ]1 Q(~-)II, are non-zero constants. The premult ipl icat ion by P(k) is equivalent to a combination of two kinds of e lementary operations on rows, and the post -mul t ip l icat ion by Q(k) is equi- valent to a combinat ion of corresponding operations on columns.

These four e lementary operations on the rows and columns of A - - ) , I correspond to simple operations on the system 1(2) wri t ten in the form

d t2) ( A - - D I ) x - - O , D = dr '

where A - - D I is considered as a polynomial in the operator D, and so the algebraic processes of finding the standard normal by polynomial matrices can be followed s t ep -by- s t ep for the system (2). From the standard normal form the JORDAn normal form can be found by the methods of § 8. For the standard normal form consists of diagonal blocks of the form B in 8 (2). In terms of (2 ) the four e lementary operations are (i) mult ipl ication of the i th equat ion by a non-zero constant t~, (it) operation on the jth equation by a polynomial operator Psi(D) and adding it to the i th equation, (iii) replacing the i ta coordinate x~ of x by ~tix{ where ~t~ is a non-zero constant, (iv) replacing x~ by x~'- 'x~+qii(D)x], where qii(D) is a polynomial operator. The first two operations obviously each correspond to algebraic operations on rows which are equivalent to premult iplying by a t r iangular polynomial matr ix P(D) with non-zero constants along the principal diagonal so that its determinant is a non-zero constant. Similarly the third and fourth operations correspond to algebraic operations on columns which are equivalent to put t ing y = Q(D)x, where Q(D) is a t r iangular polynomial matrix which non-zero elements along the principal diagonal so lhat its determinant is a non-zero constant. Any combination of such operations also corresponds to premult ipl icat ion and post-mult ipl icat ion by polynomial matrices with deter- minants which are non-zero constants, and, since this is so, such operations all have inverses. In fact in the case of (i) and {iii) the inverses are multi- plication by ~t1-1, and "in the case of (it) and (iv) the inverses are obtained by changing the sign of pit(D) and q~i(D) respectively. It is convenient to

:ft. L. CAI{'~wI¢I(xH'r: Reduction1 of ,qy.~tem.~ of Li~car t)iffcrc~,tiat, etc. 157

observe that any two rows or columns can be interchanged by repetitions of the four operations.

TI-IEORE~ 4. - By means of the four elementary operations the system (2) can be reduced to the form

{3) E(D)y -- O, y -- Q(D)x

where Q(D) is a polynomial matr ix whose determinant is a non-zero constant, and E(D) is a diagonal matrix such that the element e~fD) of the i th row and column is a polynomial in D with leading coefficient I, and edD) is a factor of e~+~iD), i - ~ 1 , 2 . . . n - - 1 . Further i f ~--) ,~ is a factor of ed), ) repeated n~,~ (~) times, where 0~_ n~,i ~_n~,~+~, i - - I, 2 , . . . n - - 1 , then ~.~ is a root of I' A - - ~Itl = 0 repeated N~ limes and Z n~,~ -- 25~.

As in {1) any combination of the four elementary operations gives a system E(D)y such that

E(D)y -- P(D)(A - - DI)Q(D)x,

where I] P(D)II, I] Q(D) II are non-zero constants. Hence if E(k) is a diagonal matr ix of the form stated,

(41 I[ Et )I[ = H e,(X) = M . Lt i - - H,

where M is a non-zero constant, and the ll~st part of the theorem follows from this.

]f A is already in JORDAN normal form, there is little point in performing the reduction of this theorem, but it should observed that A - - ; k I is not in the precise form described unless A is in diagonal form and all its elements are equal, in which case e d D ) - - D - - X , for i : 1, 2 . . .n . If A is not in dia- gonal form, there is at least one non-zero constant element, and if A is in diagonal form but the a~i are not all equal, it is easy to obtain a non-zero constant element in A - - ~I by means of operations (ii) and (iv). When a non-zero constant element has been obtained, change rows and columns until it is in the top lef t -hand corner and reduce it to 1. Then by means of the elementary operations we can reduce all the other elements of the first row and column to zero. The n - 1 by n - 1 matr ix A,_~(D) consisting of the remaining n - - 1 rows and columns is treated similarly. By means of the elementary operations we put the non-zero element of lowest degree in the upper le f t -hand corner of A,,_~(D) after reducing the elements in such a way that this element

(6) I t should be observed that ns~i may be 0 for i : l , ~ , . . . n - - 1 , and is not ns(k).

158 N . L . CAItT'WRIGHT: R e d u c t i o n of S y s t e m s of L i ~ w a r D i f f e r e n t i a l , c t e

g~(D), say, is the one of lowest degree which can be obtained by such operations. It is then a factor of all the ether elements in the same row and column. For if g~k(D) is the element of the first row nnd k th column of A, ,_ l l D) such that g~k(D) = g~ t (D)g , . lD)q~(D) + r~h(D), where qi, k(Dl and ra~(D)

are not identically zero and r~k(D) is of lower degree than g~{D), operating on the first column with --q~h(D} and adding it to the k th, we can obtain an element r~,,(D} of lower degree than g~dD). The firs~ column is dealt with similarly, and of course the leading coefficient of g n ( D ) can be reduced to 1 by operation (i) I t is then required element e~.(D).

The matr ix A,~_~(D) consisting of the last n - - 2 rows and columns is dealt with in the same way. A modification of the argument by which we showed that g~dD) is a factor of g ~ ( D ) also shows that it is a factor of M1 the elements of A~_~{D), and repeating the process we obtain the result required.

10. Standard Normal F o r m . - From 9(3) we get the standard normal form of A consisting of blocks Bi of the form of B in 8(2) along the principal diagonal and zeros elsewhere. Each block Bi corresponds to one of the elemenls ei(D} of positive degree in D. For according to theorem 4, if A is not diagonal matr ix with all its elements equal, e ~ ( D ) - 1, and so by the theorem we may suppose that ei(D) is of degree n~, where ni -- 0, i -- 1, 2, ... r, 1 --_~ *~i---~ ni+~, i -- r + l, r + 2, ... n -- 1. Then from 9(3) we have

II1 e i ( D ) y i ' ~ Yi = 0, i = 1, 2, ... r,

and the remaining equations

ei(D)yi = (D'*~ + b,iD~i -~ + ... + b,,ii)Yl = O, i = r + 1, r + 2, ... n ,

are of the form 8(1). Hence for fixed i putt ing

(:?) z i - - Y i , z~+l~ = D z i + ~ - ~ , k = 1, 2 . . . . n l - - I,

we obtain a system

zj+~;_~ - - zi+l~ , k = 1, 2, . . n l - - 1,

z i ~ "~i -1 = - - b ,~ z i - - b,~:_ 1, ~ zi+1 - - ... - - b: z j+ , , r~ ,

which can be represented by

(B~ - - Dlz(i) = O.

where z <i) is a column vector with coordinates (z i , z i+1, zi+n~-~) where i "-- j ( i ) ,

M. L. CAtc,rWl~i(ni'r: Reduction of Systems of Li,~war Differential, etc. 159

and B~ is a matrix of the and since by 9(4t

form 8(2). This holds for i = r -]- 1, r -4- 2 .... n,

Vb

n i ~ n~

by put t ing j ( r + l } = 1. j ( i + l } = j ( i ) - { - n ~ , i - - r + t , . . . n - - 1 , we obtain a vector z and n - - r blocks B, such that

(Bo }

\o °l'"l ° 1 i o

o 1 i k £

Z - " O -

The transformation z - R(D)y as formulated in (2) has ]t R I I - 0, which is

not permitted, but the total number of zi(~k, k - - 1, 2 .... n ~ - 1 is E (n~-- 1) --- r-~ 2

= n - - ( n - - r ) - - r . I-Ience by (1) we may add one of the coordinates ~1, y~, ... y,. to each of the zi:~)+k for which k ~ 1, without altering its value, and at the same time obtain a polynomial matrix with a non-zero constant deter- minant. For R is then a t r iangular matrix with 1' s along the principal dia- gonal.

An alternative way of looking at the problem is to say that since the t ransformations used to obtain E(D)y from {A ~ D I ) x all have inverses it is possible to obtain the original system from E(D)y, and, since y l - y ~ - - . . . - = y , . - - 0 , the remaining functions D'~y~, m - - O , 1, 2 , . . .n~, i - - r ~ - l , . . . n span the vector space of x and all the coordinates of x can be expressed in terms of them. It should be observed that the original system 1(2) provides a method of expressing Dx~ (and therefore D"x.i) for any integer m in terms of x~, 0c2 ... x~ and the elements of A.

As we showed in § 8 the J'OI~DAN normal form of a block B~ can be determined immediately from the roots of its characterist ic equation. Hence the polynomial operators e,(D), i - - 1 , 2, . . .n, determine the JORDAN normal form of A, and we have the following result.

THEOREM 5. - If the factor D - - ks does not occur in e~(D), i - - 1, 2, ..., n -- K(s}, and occurs repeated ns, ~ ~ 1 t imes in e~(D), i - - n - - K(s) ~ I, ... n, then the Jordan normal form of A contains K(s) blocks of the form 2{3) such that ns(k) - - n,,~, i - - n - - K(s) T 1, n -- K(s) --~ 2, .., n, and conversely.

160 M:. L. CAR'I'WI~mlt'±': Rcd,wtion of Sy.~tcms of Linear DiffcJ'cJ~tial, etc.

T h e fac to r s ( ~ - ~81".¢(k), k : 1, 2 ... K(s), s = 1, 2, ... S, a re usua l ly cal led the e l e m e n t a r y d ivisors of A - - ) . I and of all po lynomia l ma t r i c e s s imi la r to it, but SCgREIER and SPER~ER app ly this n a m e to the po lynomia l s e~{k), i - 1, 2, ... n. I n both cases if tile e l e m e n t a r y d iv isors are k n o w n the JOaDAI'~ no rma l fo rm can be wr i t t en down.

I t may be obse rved that this me thod es tab l i shes that

( J - - D I ) y - - P(D)(A -- DItQ(D)x ,

w h e r e P(D), Q(D) are sti l l po lynomia l ma t r i c e s whose d e t e r m i n a n t s are n o n - zero eos tants . F r o m this it can be shown (7) tha t the re exis ts a cons t an t

m a t r i x P such tha t

J = P A P -I,

or, as we have j u s t observed , t e rms of the fo rm Dmx~ can be exp re s sed

in t e rms of A and x, and so this gives a d i r ec t me thod of f ind ing J and y

in t e rms of A and ~e.

(7) See .L A. TODD, loe. cir. 15~-~.