S0002-9904-1930-05041-7

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i93°-l ROOTS OF A MATRIX 705

T H E C H A R A C T E R I S T I C R O O T S O F A M A T R I X *

B Y E . T . B R O WN E

1. Introduction. If A is a square matr ix of order n a n d Iis the un i t m at r ix , the equ a t ion in X obta ined by equa t ing toz e r o t h e d e t e r m i n a n t \A— \l\ is called the characteristic equa

tion of A. The roo ts o f th i s equa t ion a re ca l led the characteristic roots of A . A l tho ug h i t is no t possible to m ak e an y def ini tes ta tement as to the na ture of the charac te r i s t ic roo ts o f thegenera l a lgebra ic mat r ix A, severa l au thors have g iven upperl im its to th e ro ots . T h e f irs t up pe r l imit seems to ha ve beengiven by Bendixsonf in 1900.

Let us denote by A f a n d A t he t r anspose and the con juga t eimaginary , respec t ive ly , o f the square mat r ix A. If we write

A+T A-T

i t i s obvious tha t B'= B so t ha t B is Hermit ian (or real symm etr i c if A i s rea l ) . S imi la r ly , Cis H erm i t ian (or skew -sym m etr icif A is rea l ) . Be ndix son ' s theore m th en is as fo l lows:

B E N D I X S O N ' S T H E O R E M , (a) If a+if3 is a characteristic rootof a real matrix A and if p i^ p2 ^ • • • ^p n are the characteristicroots (all real) of the symm etric matrix B = (A ~\-A')/2, then

(1) Pl è « â Pn.

(b) If gn is the greatest of the numerical values of the elements\(aij — aji)/2\ of the real skew-symm etric m atrix (A— A')/2,

then

(2) |/3 | ikg"[n{n- l ) / 2 ] " 2 .

T h e ex tens ion to the case wh ere the e leme nts of A are complexwas mad e in 1902 by H irsch f who pro ved t he fo llowing theore m :

* Presen ted to the Soc ie ty , Sep tember 11, 1930 .t Bend ixson , Sur les racines à1 une équation fondamentale, A c t a M a t h e

mat ica , vo l . 25 (1902) , pp . 359-365 .î H i r s c h , A c t a M a t h e m a t i c a , v o l . 2 5 ( 19 0 2 ) , p p . 3 6 7 - 3 7 0 .

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706 E. T. BROWNE [October ,

H I R S C H ' S T H E O R E M , (a) If a+i/3 is a characteristic root ofany square ma trix A and if we designate by g the greatest of \a.i3- | ,*by g' the greatest of |(a t- ?-+â/ i ) /2 | , and by gn the greatest of\(dij-— a.ji)/2 |, then always

(3) | a + ifi | ^ ng,

(4) | a | ^ ng',

(5 ) \P\£ ng".

(b) If pi is //ze greatest and pn the least {algebraically) of thecharacteristic roots of B ~ {A +A ')/2, then always

(10 Pi è a ^ Pn.

In 1904 Bromwichf gave a proof of Hi rsch ' s Theorem (a) and(b) and fur ther ex tended (b) as fo l lows:

B R O M W I C H ' S T H E O R E M . If a+i/3 is a characteristic root of amatrix A and if we denote by ±/x i, • • •, ± \xv (2v^n) the nonzero characteristic roots of the matrix C— (A— A')/(2i), then |/3 |cannot exceed the greatest of the | /x; |.

In 1922, for a real matr ix A, Pickf gave a proof of Bendix-son ' s Theo rem (a ) w i th Bromwich ' s ex t ens ion and he showedth a t ( 2 ) can be rep laced by |/3 | =g tr | c tn ir/(2n) | wh ich in genera lg ives a more res t r ic ted l imi t than (2) .

In 1927 the au th or § a t tac ke d th e pro blem f rom a d i ffe ren tang le and p roved the fo l lowing theo rem:

If X i s a ch arac te r i s t ic roo t o f a squa re m at r ix A and ifpi^p2^ • • 'z^Pn are the charac te r i s t ic roo ts (a l l ^0) o f AA',t h e n p i ^ X X ^ P n -

I t i s the purpose of th i s paper to show by a very s implemethod tha t Hi rsch ' s l imi t s (3 ) , (4) and (5) may be rep laced

* H e r e \A n | d e n o t e s n o t t h e d e t e r m i n a n t o f t h e m a t r i x A b u t t h e a b s o l u t ev a l u e o f t h e n u m b e r An,

t B r o m w i c h , On the roots of the characteristic equation of a linear substitution, A c t a M a t h e m a t i c a , v o l . 3 0 ( 19 0 6 ) , p p . 2 9 5 - 3 0 4 .

Î P ick , Über die Wurzeln der charakteristischen Gleichung von Schwing-ungsproblemen, Z e i t s c h r i f t f u r a n g e w a n d t e M a t h e m a t i k u n d M e c h a n i k , v o l .2 ( 19 2 2 ) , p p . 3 5 3 - 3 5 7 .

§ The characteristic equation of a matrix, th i s Bu l le t in , vo l . 34 (1928) , pp .3 6 3 - 3 6 8 .

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193°-] R O O T S O F A M A T R I X 707

by l imi t s which never exceed the former and a re in genera lmore r e s t r i c t ed .

2. The Characteristic Roots of A A'. HA i s any squarema t r ix , r ea l o r complex , Au tonne* has shown tha t t he r e ex i s ttwo un i ta ry ( real o r th og on al , if A is rea l ) m at r ice s P , Q s u c h t h a t

(6) A = P'NQ

w h e r e N has rea l pos i t ive (or zero) numbers in the main d i

agona l and ze ros e l s ewhere . On fo rming the p ro du c t A A'= P'N2P i t becomes ev iden t t ha t t he number s i n t he d i agona lof N are the pos i t ive (or zero) square roo ts p i 1 /2 , • • •, p n

1 / 2 ofthe charac te r i s t ic roo ts o f A A ' . F r om (6) , N = PAQ', s o t h a ti f we denote by pa, tin a n d qa the e lement in the ith row andt h e jth co lumn o f P , N a n d Q, r e spec t ive ly , we have

\1 ) Mij ~ / jPir&rsQis «r ,s

Tha t i s , t he e l emen t s o f N are of the form X *'i' aaxiy j w h e r e(xi, • • • ,x n) a n d (yi, • • • ,y n) a re s e t s o f number s such tha t/ .i^iXi

=/ jjy%yi^ i •

3. An Upper Limit to the Roots of A. Le t u s deno te by rji a n df i the abso lu te va lues of x% a n d y^ r e spec t ive ly . Then the s e t s(yi, • ' ' > V n) a n d (fi, • • •, fn) a re rea l se t s such tha t YjVi2 = X ) f *2

= 1. Since rj i and f* are real

(8) mïiÛîM + t f ) .

H e n c e

|»« I = I ^aaxiyj | ^ X ) I aa I I xi I I yi I

= X) I Ö»J I rati < i X I öi,-1 (v? + f ? ) .

I f 5 i denotes the sum of the abso lu te va lues of the e lements inth e ith row of A and if S i s the grea tes t o f the S i} w e h a v e

* A u t o n n e , Sur les matrices hy•/oherm itiennes et les unitaires, C o m p t e sR e n d u s , v o l . 15 6 ( 19 13 ) , p p . 8 5 8 - 8 6 0 , i n w h i c h t h e t h e o r e m i s g i v e n w i t h o u tproof. I n f ac t t h e t h e o r e m f ol lo w s a s a c o n s e q u e n c e of t h e a u t h o r ' s T h e o r e mIV on p . 367 of th e a fo reme nt ione d pap er . See a l so Ta be r , On the linear transformations between two quadrics, P r o c e e d i n g s o f t h e L o n d o n M a t h e m a t i c a lSoc ie ty , vo l . 24 (1892-93) , pp . 290-306 , in which the theorem fo r A rea l andn o n - s i n g u l a r is c o n t a i n e d i m p l i c i t l y .

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708 E. T. BROWNE [October ,

(9) X ) I <*u I V i2 = Jlvi Z ) I aa I = Z ) ^ i 2 ^ i = 5 Hi? = 5 -•i , i i i i

Similar ly , i f Ti denotes the sum of the abso lu te va lues of thee lements in the i th co lumn of A and if T is the greatest ofth e Tu

£ | a < / l * 7 ^ T.

H e n c e

| » « | = Pim ^ (S + T)/2.Now f rom the au thor ' s theorem as quoted ear l ie r in the paper ,if X is a ch ara cte r is t i c roo t of A a n d M is the greatest of thep's, t h e n

| X | ^ M 11 2.

H e n c e

| X | ^ ( 5 + T)/2.

We therefore have the fo l lowing theorem:T H E O R E M . If Si (Ti) is the sum of the absolute values of the

elements in the ith row (column) of a square matrix A and if S (T)is the greatest of the Si (Ti), the absolute va lue |X | of a characteristic root X of A cannot exceed (S+T)/2.

Equat ion (3) o f Hi rsch ' s Theorem (a) obvious ly fo l lows as acoro l lary to this th eo rem . I t is c lear th a t the l imit given by th ela t te r can never exceed tha t g iven by Hirsch ' s c r i te r ion and i s

in general less .The l imi t i s somet imes ac tua l ly a t ta ined , fo r example , i f A

is a matr ix each of whose elements is the square of the correspo ndin g e lem ent of a rea l o r tho gon al ma t r ix . In th i s case Aobv ious ly has t he cha rac t e r i s t i c r oo t + 1 . M oreove r , t he lim i ti s a lways a t ta ined i f A is a circulant, ( tha t i s , aij = aj-i+i, fo r i^j;aij = an+ j-i+i for i>j) whose e l emen t s a r e r ea l and ^0 .

In par t icu la r , i f A i s He rmi t i an , Si—Tz and hence S=T,so tha t we have the fo l lowing coro l la ry .

C O R O L L A R Y 1. If Si is the sum of the absolute values of theelements in the ith row of an Herm itian matrix A, and if S is thegreatest of the Si, the numerical value of a characteristic root X ofA cannot exceed S.

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I930-] ROOTS OF A MATRIX 709

The l imi t g iven in the coro l la ry i s ev ident ly a t ta ined i f Acon sis ts ent i re ly of zeros exc ept in th e diag ona l . I t is a lso atta ined i f A is a real cyclic mat r ix , ( t ha t i s , aij = ak where k isthe least posi t ive residue of i+j—1, m od n) with posi t ive orze ro e l emen t s .

On m akin g use of H i rsch ' s The ore m (b) we hav e a lso th efol lowing corol lary.

C O R O L L A R Y 2. If a +i/3 is a chara cteristic root of A and if S/

is the sum of the absolute values of the elements in the jth row ofthe Herm itian matrix B — (A-\-A ')/2, then if S' is the greatestof the S/, it follows that \a \ ^S'.

Equat ion (4) o f Hi rsch ' s Theorem fo l lows d i rec t ly f rom th iscor ol lary . M ore ov er , i t i s c lear th a t our cr i ter ion usua l ly givesa nar rower l imi t than tha t o f Hi rsch .

By invoking Bromwich ' s Theorem we can s ta te a l so the fo l lowing coro l la ry .

C O R O L L A R Y 3 . If a+ift is a chara cteristic root of A and if S/'is the sum of the absolute values of the elements in the jth row ofthe Hermitian matrix C= (A — A r)/(2i), then if Sn is the greatestof the S/', we have \j3 | ^S".

Equat ion (5) of Hirsch 's Theorem (a) fol lows direct ly f romthis coro l la ry . Coro l la r ies 2 and 3 , which h ave been dedu cedf rom ou r t heo rem, migh t have been p roved d i r ec t ly w i thou tinvo king Au ton ne ' s the ore m . Fo r if we wr i te , as in §1 , B —

{A + 2 " 0 / 2 , C= (A -1f

)/{2i), i t is c lear th a t A =B+iC w h e r e Ba n d C a re H erm i t i an m a t r i ce s . Suppose now th a t a-\-i/3 is acharac te r i s t ic roo t o f A . T he n the re exis ts a set (xi, • • • , xn)^ (0, • • • , 0) , an d wh ich we m ay su ppo se to ha ve been divide dth rough by the p rope r non -van i sh ing f ac to r so t ha t ^XiXi=l,s u c h t h a t

^2atjXj = ^btjXj + i ^CtjXj = (a + i$)x t it = 1, • • • , n).i i i

On mul t i p ly ing the se equa t ions t h rough by x t a n d s u m m i n gas to t, we have

(10) ^btjXtXj + i ^CtjXtXj = X )(a + i(3)x tx t = a + if3.t,j t,j t

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710 J . V. USPENSKY [October,

Now s ince B a n d C a r e He r mi t i a n ma t r i c e s e a c h o f t h e s u mma t ions on th e l ef t i s r ea l . H ence , equ a t in g rea l and im ag in arypar t s in (10) we have*

^" ' J 'ii%i%'it

i* 3

P ' / jCi i%'i% i »

i,j

I f now we deno te by rji the abso lu te va lue o f Xi and p roceed

as in (9) , Corol lar ies 2 and 3 fo l low at once .THE UNIVERSITY OF NORTH CAROLINA

O N T H E R E D U C T I O N O F T H E I N D E F I N I T EB I N A R Y Q U A D R A T I C F O R M S f

B Y J . V . U S P E N S K Y

The reduc t ion theory o f the indef in i t e b ina ry fo rms has beenpresented in widely d i f ferent forms and f rom var ious points ofv iew. B u t w ha tev er po in t of v iew is ad op ted , i t seems th a tHermi te ' s p r inc ip le o f con t inuous va r iab les under more o r l e s sdisguised form const i tu tes an essent ia l foundat ion of a l l theex i s t ing theor ies o f r educ t ion .

Hermite 's pr inciple in i t s s imples t aspect consis ts in associa t ion wi th a g iven indef in i te form of a posi t ive quadrat ic form

c o n t a i n i n g a c o n t i n u o u s l y v a r y i n g p o s i t i v e p a r a me t e r a n d t h es tudy of in tegra l values of var iables which give success ivem i n i m a of t h e l a t t e r . Ho we v e r , t h e r e d u c e d fo r ms in H e r m i t e ' stheory d i f fer f rom those in the c lass ica l Gauss ian theory of redu c t ion . A l i t t l e con t r ibu t ion to the theory of r educ t ion wh ichth i s a r t i c le con ta ins has fo r i t s purpose to show how, by subs t i t u t i n g f o r He r mi t e ' s p o s i t i v e q u a d r a t i c f o r m a c e r t a i n n o n -h o mo g e n e o u s f u n c t i o n c o n t a i n i n g a v a r i a b l e p o s i t i v e p a r a mete r , we ob ta in p rec i se ly the Gauss ian reduced fo rms .

L e t £ = ax + fiy, rj = yx + 8y

* See Hirsch, loc . c i t . , p . 369.f P resen ted to the Soc ie ty , Apr i l 5 , 1930 .