Theorem application to companion matrix of polynomia

Painter, R. J.

English

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& . e\ . . 
, FINAL REPORT 
t o  
Nat ional  Aeronautics and Space Adminis t ra t ion 
f o r  
Research Grant Number SC-NGR-06-002-041 
Bo nds for t h e  Zeros of Polynomials 
" Y 
B N_6 7 - 40494 
(ACCESSION UMBER) 
I 14 
m 
z e I / / .  I - -  . 
(CAGES) 
(NASA CR OR TMX OR ADAUMBER) 
g 
P 
i u : 
P r i n c i p a l  I n v e s t i g a t o r :  Richard J .  P a i n t e r  
P r o j e c t  Per iod:  1 A p r i l  1 9 6 6  - 30 September 1967 
I /  Department of Mathematics and S t a t i s t i c s -  
Colorado S t a t e  Un ive r s i ty  
Fort C o l l i n s ,  Colorado 
1 October 1967 . . .  . 
M 
https://ntrs.nasa.gov/search.jsp?R=19670031165 2020-03-16T16:42:16+00:00Z
. 
The fo l lowing  r e s u l t s  are der ived from an a p p l i c a t i o n  of a theorem 
of A .  Brauer t o  t h e  companion matr ix  of a polynomial and a p p l i c a t i o n  o f  t h e  
theorem of Rouche. The p a t t e r n  fol lows t h a t  of  t h e  theorems mentioned i n  
t h e  c o n t r a c t  p roposa l  and a r e  i n  some sense a rounding out  of  t hose  r e s u l t s  
der ived  e a r l i e r  by t h e  p r i n c i p a l  i n v e s t i g a t o r .  No negat ive  r e s u l t s  (and 
t h e r e  were many) are r e p o r t e d ,  
It was suggested i n  t h e  f i r s t  progress  r e p o r t  t h a t  t h e  r e s u l t s  o f  
t h i s  s tudy ,  and more l i k e l y  t h e  study i t s e l f ,  may have a p p l i c a t i o n s  i n  
ob ta in ing  a n a l y t i c  c r i t e r i a  for i r r e d u c i b i l i t y  of polynomials wi th  i n t e g e r  
c o e f f i c i e n t s .  
undertaken s h o r t l y  by t h e  p r i n c i p a l  i n v e s t i g a t o r .  
w i l l  be a p p r o p r i a t e l y  c r e d i t e d  t o  t h i s  g r a n t .  
This  has no t  been pursued under t h i s  g r a n t ,  but  w i l l  be 
Any r e s u l t s  publ i shed  
The p r i n c i p a l  i n v e s t i g a t o r  wishes t o  t a k e  t h i s  oppor tuni ty  t o  thank 
t h e  Nat iona l  Aeronaut ics  and Space A d m i n i s t r a t i m  for t h e  oppor tun i ty  
a f fo rded  by t h i s  g r a n t  t o  pursue these s t u d i e s .  
- 2 -  
I n  el] , Brauer proves t h e  fol lowing 
Theorem. 
e n t r i e s .  For each K and X with K # X one s e t s  
Suppose A = ( a Y h )  i s  a square ma t r ix  o f  o r d e r  n with complex 
I )  Q,, I - C{ la ,pJ  + IaX,alcvl + laKvaXv 
V 
where and v run  from 1 t o  n but  P f K,X and v # K,X. Moreover, l e t  
denote  t h e  p r i n c i p a l  subdeterminant of t h e  c h a r a c t e r i s t i c  determinant  
of  A which c o n t a i n s  t h e  rows K and A ,  Then each c h a r a c t e r i s t i c  r o o t  of A 
D K X  
l i e s  i n  t h e  union o f  t h e  C a s s i n i  ova l s  
We wish t o  apply t h i s  theorem t o  t h e  companion matrix 
-a 
3 -a n-1 n -a 2 -a 1 -a ([  ; 0 1 ; i : ;  ;) 
. e *  
n n-1 + 
o f  t h e  polynomial f = z t a z t a e We first note  t h a t  1 n 
= (w-a ) ( w - a  ) - a 
K x ~ x K  D K X  K K  AX 
and observe t h a t  for t h e  above companion ma t r ix  a l l  d i agona l  elements are 
z e r o  except  i n  t h e  1,l p o s i t i o n .  Thus i f  K = 1, then  D = (w-all)w - alXaX1. l h  
Moreover , 
F u r t h e r  a 11 = -a 1 so  t h a t  t h i s  becomes DIX = (w+al)w - alXaAl' 
s i n c e  K # A ,  t h e n  X 2 2 and i n  t h e  f i rs t  column a l l  e n t r i e s  are 0 except - 
, ' .  
- 3 -  
t h e  f i rs t  two. Thus w e  have two s i t u a t i o n s :  A = 2 ,  and X > 2 .  
2 for A = 2 D12 ( w + a  > w  t a 
DIX = (wtal)w 
1 
f o r  A > 2 .  
For  K > 1 e i t h e r  a = 0 or a = 0 s i n c e  i n  t h e  companion ma t r ix  
KA A K  
t h e  only non-zero elements no t  i n  t h e  f i rs t  row occur  i n  t h e  K , K - ~  
p o s i t i o n .  Hence f o r  K > 1 w e  have only one form for D v i z . ,  
KX ' 
DKA w 2 *  
We now wish t o  c a l c u l a t e  t h e  Q va lues  for our  d i s t i n c t  o v a l s ,  
K A  
Q,,, QlA(A>2),  and Q K A b l ) .  
n 
v=3 
Now Zla12a2v I = la121Cla2vl = ( - a2 [  But a l l  elements o f  t h e  2nd 
V V 
row of t h e  companion ma t r ix  a r e  zero except i n  t h e  first column so  t h a t  
lla12a2vl = I-a21 1 
V v=3 
= 0 .  In  a similar manner we see t h a t  
la5[  + ... + l an [ .  NOW 
must always be zero s i n c e  a = 0 f o r  a l l  p > 1, and 1 i s  
1 v a 2 p  2u 
2va1p 
The t e r m  a 
excluded from t h e  range o f  p. 
u n l e s s  a i s  z e r o  and a 
is  not n e c e s s a r i l y  ze ro  
Since 1 i s  no t  i n  t h e  range 
The term a 
= 0 f o r  a l l  A > 1. 
2v 2v  
for X w e  have t h a t  1 lalva2p + a2valp I = 0. 
l aS/  + ... + lanl. 
Hence Q,, = la3 I + la4 I + 
V - W  
Consider now Q f o r  X > 2. 1X 
- 4 -  
Now by examining the companion matrix we easily see that 1 (alXaAv I = laA 1 , 
V 
. . .  0 D . C . . O .  
t 
+ a  I +  
V<A -1 
I +  
v<x t1 
1 lalvaX,A+2 + aAval,A+2 I +  
. 0 . 0 0 . 0 . . . . . . .  
+ 
1 IalvaAn + aAvalnl9 v<n 
run through the admissible values for p .  
= 0 for pi < A -1 and a 
Now each sum in 1 4 2  where p 
(A) is zero, since a 
In the sum (B) we have a 
= 0 f o r  v < p < A -1. 
hi A V  i 
= 1 but a = 0 for all v < X -1 so that 
A , A - 1  AV 
A-2 
i=2 the sum ( B )  is 1 /ail. In the sums in ( C )  we have a = O a n d a  = O  A V  A , A + j  
except for v A - 1, so that each sum collapses to a single term 
- 5 -  
n 
I .  Hence t h e  sum of all sums i n  ( C >  is 1 lai I .  Therefore  
:=At1 
lal,A+j 
i#A -1, A 
We now c a l c u l a t e  Q K A  for K > 1. 
F i r s t  cons ider  t h e  s p e c i a l  case  when A = 1, and examine t h e  sums one a t  a 
t i m e  . 
Moreover 
= i ,lKI 
f o r  K z 2  
f o r  ~ > 2 ,  
f o r  ~ = 2  
f o r  ~ = 2  
f o r  K > 2 .  
if1 ,r-i , K 
- 6 -  
2 Hence Q 
Now we c a l c u l a t e  Q for K , A  > 1. We have 
KX 
c laKvaXv I 0 
V 
Hence f o r  K , A  > 1 w e  have 
1 
Q K X  = 1 1  
2 
SYNPOSIS: The d i s t i n c t  o v a l s  t hen  a r e  
(A 1 Iw(w+al) + a21 2 la3[ + laql + ... + lanl 
for X = K - ~  
for A#r-l, 
f o r  X = K + ~  
for A # K + 1 .  
for h = ~ - l  
f o r  X = K + 1  
o the rwise  
for X = K - ~  
f o r  X = K + 1  
otherwise 
= 1 or  2 according as A = K ?  1 o r  n o t .  
Q K X  
n 
1 (ail was denoted by S and t h e r e  
i = 2  
I n  t h e  c o n t r a c t  p roposa l  t h e  sum 
w e  o u t l i n e d  t h e  ea r l i e r  work of M. Parodi  and of t h e  p r i n c i p a l  i n v e s t i -  
g a t o r  i n  d e a l i n g  wi th  t h e  c i r c l e  
I w l  =" 4-5- 
and wi th  t h e  o v a l  (w(wta l ) l5  - S. I n  t h e  earlier work simple bounds were 
- I -  
obta ined  by s u b j e c t i n g  t h e  companion mat r ix  t o  c e r t a i n  s i m i l a r i t y  t r ans fo rma t ions ,  
But i n  t h e  p re sen t  s i t u a t i o n  t h e  oval  (B) i s  conta ined  i n  t h e  ova l  ( A ) ,  s i n c e  
and t h e  ear l ie r  techniques  seem not  t o  y i e l d  f r u i t f u l  r e s u l t s  here .  
may, however, a s k  under what c i rcumstances t h e  ova l  (A) may be used f o r  
l o c a l i z a t i o n  o f  t h e  ze ros  o f  f ,  and t h i s  i s  t h e  s u b j e c t  o f  t h e  fo l lowing  
remarks .  
One 
Consider t h e  Cass in i  ova l  
Q :  1 w 2 + a w + a  I < K  
1 2 =  
whose f o c i  f and f are g iven  by 1 2 
f l  $-al + E}  and 1 2  
f = -{-a 1 - d q z  
2 2 1  2 '  
Thus Q i s  
I w  - f l (  Iw - f I < K. 2 =  
2 L e t  u s  cons ide r  t h e  r e s t r i c t e d  s i t u a t i o n  where t h e  c o e f f i c i e n t s  a 
are real  and a2 2 4a2. 
midpoint o f  t h e  l i n e  segment jo in ing  t h e  f o c i  does not  belong t o  $2; t h a t  i s ,  
provided 
and a 1 
Now t h e  oval  s p l i t s  i n t o  two branches provided t h e  
f i g .  l a  
al>O, a > o  2 
f i g .  l b  
al<O a > O  2 
- 8 -  
f l t f 2  f l t f 2  
-- 2 f l  I /  2- f 2  
f -f l 2  > 4K f-) 1 2  
\${-al t -1 - ~ - a l  1 -da2-4a2}I2 > 4K e 1 
Moreover, if t h e  o v a l  Q i s  disconnected,  t hen  t h e  p o i n t  B w i th  minimum 
modulus on t h e  branch con ta in ing  f i n  t h e  case  of f i g .  l a  (or c o n t a i n i n g  2 
f i n  t h e  case i l l u s t r a t e d  i n  Figure lb) has modulus ( s e e  2 Or 3 ) 1 
Hence IBI 2 1 i f  and only i f  - 
lal] + 4a2-4a -4K 2 2.  - 1 2  
If w e  l e t  S '  denote t h e  sum la 1 t la4(  t ..- + ( an land  Q 
of R con ta in ing  f 2  t h e n  w e  may formulate  t h e  fol lowing 
Theorem 1. Suppose f = z + a z t . o .  t a has t h e  p rope r ty  t h a t  al > 0 ,  1 n 
a and K is  any real  number such t h a t  
denote  t h e  branch 
f2 3 
n n-1 
is rea l ,  a2 > 4a 2 1 =  2 
a2 - 4a > 16K - 1 6 s '  1 2 =  - i> 
ii 1 
Then t h e  o v a l  Q: Iw(wtal) t a2 I 5 K is disconnected and Q Contains 
f r l  
L 
e x a c t l y  one ze ro  of f. 
P r o o f :  
on t h e  boundary of Q 
The preceding d i s c u s s i o n  and t h e  hypothesis  a s s u r e  t h a t  for each w 
The hypo thes i s  gua ran tees  K > S' , w e  have \ w \  2 1. - 
f , 
L 
so  t h a t  f o r  each w on t h e  boundary o f  R 
f 2  
- 9 -  
1wn + a w n-1 + a w n-2 I = lw2 t a w t a 1 1 w  n-2 I lw2  t a w + a 1 l w ~ - ~ 1  
1 2 1 2 1 2 
> K lwn-31 > SI lwn-31 = n 1 l a . l w n - 3 1  
- 1 - i = 3  
w n- i I 2 I n 1 a iwn- i l .  
i = 3  
n n-2 Hence f and w t a w n - l  t a w 
1 2 
by t h e  theorem of R o u c d .  
have t h e  same number of ze ros  i n t e r i o r  t o  
But t h e  midpoint of t h e  l i n e  segment j o i n i n g  
t h e  f o c i  is -a / 2  so  t h a t  i n  t h e  case under c o n s i d e r a t i o n  t h e  o r i g i n  is not  
n-2 
i n  52 . Hence f i s  t h e  on ly  zero o f  wn + a w i n  0 and t h e  theorem 
fo l lows .  
52f2 
1 
f 2  f 2  2 1 
A s imilar  theorem ho lds  i n  t h e  case where al < 0 (see f i g .  lb) for 
1' t h e  branch 52 which c o n t a i n s  f fl 
We cont inue t o  assume a > 0 and cons ide r  t h e  s i t u a t i o n  f o r  a2 > 0 .  1 
Suppose t h a t  t h e  ova l  52 is 
f 
disconnected and t h a t  t h e  branch il con ta in ing  
fl 
a l s o  c o n t a i n s  t h e  o r i g i n  as i n  f i g u r e  2 ,  1 
f i g .  2 
I t  i s  known ( s e e  f 2 1  or [3j 1 t h a t  t h e  c i r c l e  with c e n t e r  f and r a d i u s  Ifl - c I  1 
i s  con ta ined  i n  R 
Now 
; hence i f  I C  I 2 I t h e n  t h e  u n i t  c i r c l e  a l s o  l i e s  i n  52 . - 
fl fl 
r .  
- 10 - 
so t h a t  I C /  1 1 if and only i f  Jaf-4a2t4k - a > 2 ;  t h a t  is ,  i f  and only i f  - 1 =  
t a  t 1 .  
We are now a b l e  t o  formulate  t h e  fol lowing 
K 2 a1 2 
Theorem 2 .  Suppose t h a t  a a > 0 ,  t h a t  a: - 4a > 0 and t h a t  K i s  any 
number such t h a t  
1, 2 2 =  
K z a  + a  t1 .  2 ii 1 - 1  
Then t h e  o v a l  52 i s  disconnected and t h e  branch 52 c o n t a i n s  e x a c t l y  n - 1 
fl 
n n-1 z e r o s  of f = z t a z + + a . 1 n 
Proof:  The hypothesis  a s s u r e s  t h a t  52 is  disconnected,  t h a t  t h e  branch 
R c o n t a i n s  t h e  u n i t  c i r c l e ,  and t h a t  fl and C a r e  sepa ra t ed  by t h e  o r i g i n  
fl 
on t h e  r e a l  a x i s ,  F i n a l l y  f o r  each w on t h e  boundary o f  R 
fl 
n- 2 
t a w I = l w 2  t a w + a 2 / l w n - 2 1  n-1 
2 1 Iwn + a w 1 
n n-1 n- 2 / Hence w + a w t a w 
z e r o s  i n t e r i o r  t o  R , namely n - 1 zeros.  
and f have, by Rouche's theorem t h e  same number o f  1 2 
fl 
Again, an  analogous r e s u l t  i s  a v a i l a b l e  t o  us  i n  t h e  case t h a t  a < 0 
1 
and a > 0 (see f i g .  lb). 
2 
, ' *  
- 11 - 
I n  t h e  fol lowing w e  l e t  B '  denote t h e  maximum of la I , la4 I , . . , l a n / .  3 
Theorem 3. Suppose t h a t  al,a2 > 0 and a: 2 4a2 and f u r t h e r  t h a t  K is any number 
such t h a t  
a2 > 16K 1 i> 
a t da2-4a2-4K > 2 1 1 ii 1 
< K. i i i )  B '  
?{a: 1 + Jaa-4a -4K) - 1 1 2  
Then t h e  C a s s i n i  oval  
i s  d isconnec ted  and t h e  branch Rf conta in ing  f con ta ins  e x a c t l y  one ze ro  2 
2 
n n-1 
O f f = z  + a z  + + a . 1 n 
Proof:  A s  i n  Theorem 1, t h e  hypothesis  ( i )  i s  s u f f i c i e n t  for  t h e  ova l  t o  be 
d isconnec ted ;  ( i i )  a s s u r e s ,  as i n  f i g u r e  l a ,  t h a t  t h e  minimum modulus on t h e  
boundary of Rf i s  g r e a t e r  t han  i,so t h a t  with (iii) w e  are  assured  t h a t  for 
2 
each w on t h e  boundary of Rf 
2 
B '  < K. 
, w e  have Thus €or each w on t h e  boundary o f  R 
f 2  
n - i  n 
i = 3  
n n 
i = 3  i = 3  
n - i  
I b i W  I ;  c 
n- 2 
= ~w n + a w n - l  + a 2 w  1 1 ,  
/ 
and t h e  theorem fo l lows  from Rouche's theorem. 
> O  and a2 > 4a and f u r t h e r  t h a t  K i s  any number 
1 @2 1 =  2 Theorem 4. Suppose t h a t  a 
such t h a t  
- .  
- 12 - 
a2 > 16K 1 i )  
K > a  + a  + 1  
1 2 ii 1 
< K. iii) 6 '  
1 
2 1  1 2  -{a 
+ Ja2-4a -4;) - 1 
Then t h e  ova l  52 is disconnected and the  branch i2 con ta ins  e x a c t l y  n - 1 
f , I 
ze ros  of  f .  
Proof:  The hypothes is  ( i) a s s u r e s  us  t h a t  52 is disconnected ;  ( i i )  a s s u r e s  
us  ( s e e  Theorem 2 )  t h a t  t h e  u n i t  c i r c l e  l i e s  i n s i d e  52 and (iii) a s s u r e s  u s  
fl 
t h a t  t h e  i n e q u a l i t y  appear ing  i n  t h e  proof of Theorem 3 i s  v a l i d  for each 
w on t h e  boundary of R e Ilence t h e  theorem follows from Rouche's / theorem. 
fl 
Literature Cited  
1. Brauer,  A .  "Uber d i e  Lage d e r  c h a r a k t e r i s t i s c h e n  Wurzeln e i n e r  Matrix," 
J. f u r  d i e  Reine und Angew. Math., Band 192,  Heft 2 ,  pp. 113-116 
(19531, 
- ----- -
Ganguli ,  S, The Theory of Higher Plane Curves, Un ive r s i ty  of  C a l c u t t a ,  - - 2. 
1926 
P a i n t e r ,  R .  Extensions of Theorems of Ostrowski on t h e  Zeros of  ---- - - 3. 
C e r t a i n  Classes of Polynomials, d o c t o r a l  d i s s e r t a t i o n ,  Univ. 
of North Caro l ina ,  Chapel Hill, N .  C . ,  1963. 
-


Bounds for the zeros of polynomials Final report, 1 Apr. 1966 - 30 Sep. 1967

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