Spectral proferties of some matrices.

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A L E K S A N D E R G R Y T C Z U K A N D M A R E K S Z A L K O V S K I

S F E C T R A L P R O P E R T I E S OF S O M E M A T R I C E S

A B S T R A C T : In this paper we s h o w s o m e spectral properties of matrices. Among o t h e r s w e prove s o m e inequalities for the characteristic roots of matrices satisfying s o m e conditions. We also give a new proof for a theorem of SzuIc.

In 1984 N.V. K u h a r e n k o 123 proved the f o l l o w i n g s p e c t r a l p r o p e r t y of nxn real m a t r i x A.

T H E O R E M A CN.W. K U H A R E N K O ) . Let A = (a. .) be an nxn r e a l V ,1

m a t r i x and let

Tr A = 0 and A_ = 5 [a..a..-a. .a..1 > 0.

2 ^ l i i j j W J O

1 £ i < j £ r,

T h e n t h e r e e x i s t s at least one p a i r of c o m p l e x - c o n j u g a t e e i g e n — v a l u e s

= ck * i dk of A s u c h t h a t

* +

I

A

, •

k k n 2

T h i s t h e o r e m h a s a p p l i c a t i o n in the theory o f d y n a m i c a l s y s t e m s .

In 1 9 8 8 T . S z u l c 131 g a v e the f o l l o w i n g g e n e r a l i z a t i o n o f T h e o r e m A.

T H E O R E M B. Let A = Ca. .) be nxn real matrix s u c h that

«- J T r2 A < -^t- A„ n—1 2

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Then there exists at least o n e pair of c o m p l e x — c o n j u g a t e e i g e n - v a l u e s

=

c

k

± i d

k

of A such that

Ci. 1)

Moreover if CI. 2 )

then

2 n A2 - C n — 1 > T r2A nCn—1>

2nA_ - C n - D T r A

Tr A £ 2 A,

if n is odd

if n is even

CI. 3 ) d. £

H

2 A2 - Tr AI if n is odd 2 A2 - Tr A if n is even

In the present p a p e r we give another p r o o f of T h e o r e m B.

Moreover we prove t h e following theorems:

THEOREM 1. Let A e M O , where H (Z) d e n o t e the set of all n * n

nxn m a t r i c e s over Z a n d let A be n o n - s i n g u l a r matrix. The necessary and s u f f i c i e n t c o n d i t i o n for , j=l,2,...,n to be a roots of unity is |X^ |=1 for j=l,2, . . . ,n.

THEOREM 2. Let A = C a . . ) be an n*n c o m p l e x matrix with

>- J

|det AI>1 and let = max |A. | w h e r e A. are the

l S j S n J J

characteristic r o o t s o f A for j = l , 2 , . . . , n , then

> 1 + i°s(det A» .

THEOREM 3. Let |X"| = max |A.| where A. for j = l , 2 , . . . , n

l ^ j S n J J

are characteristic r o o t s of A M CZ>. n £ 2 and let the _n characteristic p o l y n o m i a l of A be i r r e d u c i b l e over Z. If for

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s o m e j=l,2,. . . ,n t h e root X. is not a root o f unity t h e n ixri > 1 + f s s ^

o

where c ^ 2 is some real n u m b e r and Nq depend o n l y on n and c.

We r e m a r k that f r o m T h e o r e m 2 it f o l l o w s i m m e d i a t e l y : C O R O L L A R Y 1. Let A = Ca. .) € M C2), n 2 2 and det A * ± 1.

I J N Then

\X\ > 1 + isg-2. .

From C o r o l l a r y 1 it f o l l o w s that a c o n j e c t u r e of S c h i n z e l and Z a s s e n h a u s [23 is t r u e for all m a t r i c e s A e M ( 2 ) w i t h

n det A M ± l.

P R O O F OF T H E O R E M 1. It is easy to s e e that if X™ = 1 f o r j=l, 2, . . . , n then S u p p o s e that |\.|=1 f o r j = l , 2 , . . . „ n and let

«

C I ) f C X ) = d e t C X I - A )

be the c h a r a c t e r i s t i c p o l y n o m i a l of A. Then w e h a v e C 2 ) f C X ) = Xn + A X1 n n _ 1 + ... + A where A. <= Z and A = det A * 0. S i n c e |X. 1=1 f o r

t n J j = l , 2 , . . . , n t h u s by f o r m u l a s of V i e t e it f o l l o w s

C3> I V S ( Í j for k = l , 2 , . . . , n .

From C 3 ) it f o l l o w s t h a t t h e r e e x i s t only f i n i t e n u m b e r of p o l y n o m i a l s with i n t e g e r c o e f f i c i e n t s such t h a t

|=1 f o r j=l,2, . . . ,n.

C o n s i d e r t h e f o l l o w i n g s e q u e n c e

C I ) •• • + A ;m 5Xn _ 1 + . .

where e 2. T h e s e q u e n c e C 4 ) h a s only f i n i t e n u m b e r of d i s t i n c t e l e m e n t s , b e c a u s e

iVf I = U j Im = 1 and IAj m 5 I ^ (jj) ; j , k = l , 2 , . . , n.

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T h e r e f o r e we can take a s u b s e q u e n c e -{f C X ) r s u c h t h a t

C S ) f C X ) = f C X ) = f c x ) - ...

m_ m _ m O 1 2

where rn < tn < in < . . . . From C 4 ) and C S ) we g e t o i 2

m. m m^ m^ m ^

C 6 ) X1 = xa C l ) ' X2 = X0 C 2 J ' ••• ' = ^CTtnJ

where o C l ) , . o C n ) d e n o t e s some p e r m u t a t i o n o f . T h e r e a r e i n f i n i t e l y many e x p o n e n t s nr s u c h that mt < m2 < . . . . O n the o t h e r hand t h e r e e x i s t s o n l y n!

p e r m u t a t i o n s of the s e t ,2, . . . ,n^. T h e r e f o r e t h e r e a r e e x p o n e n t s m. and m. f o r which

m. m, m. ri>. m. m.

C 7 ) xt 1 = xt 2 , x2 1 = x2 2 , ... , xn « xn

From C 7 ) we get

X ™ = 1 f o r j = l , 2 , . . . , n

where m=m. —m. and m. -m. > 0 and the p r o o f is c o m p l e t e .

PROOF OF T H E O R E M 2. L e t f C x ) b e t h e c h a r a c t e r i s t i c p o l y n o m i a l of the m a t r i x A. It i s w e l l - k n o w n t h a t

C 8 ) I V ' X J = , d e t A |-

S i n c e |Xj I 5 I XT I f o r j = l , 2 , . . . , n , t h u s by ( 8 ) it f o l l o w s C 9 )

\ X \ *

^{|det A I}^{1 / n}^.

S i n c e |det A|>1 t h u s w e have

^ log |det AI A

C I O ) |det AI = e = 1 + £ log |det A | + ...

From C I O ) w e get 1

C l l ) |det A In > 1 + i- log Idet A | . From C l l ) and C 9 ) we o b t a i n

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)X\ > 1 + á g i d é t A j

and t h e p r o o f is c o m p l e t e .

P R O O F O F T H E O R E M 3. L e t Nq d e n o t e s t h e s e t o f a l l r o o t s o f the c h a r a c t e r i s t i c p o l y n o m i a l s of fixed d e g r e e n £ 2 w i t h integer c o e f f i c i e n t s and i r r e d u c i b l e o v e r Z such t h a t

|X| ^ c, w h e r e c £ 2 is some r e a l number. S u p p o s e that f o r every j = l , 2 , . . . , n w e have

1 ( 1 2 ) IX.I S c °

J From ( 1 2 ) we get

k . . ÍTfT*

C 1 3 ) |X* I S |X. R ^ c ° <: c

for k = l , 2 , . . . , 1 + N . From ( 1 3 ) and d e f i n i t i o n of N we get

o o

that t h e r e e x i s t s kc j 5 * k^ '5 such t h a t k <j 5, k <j 5 e

1 2 1 ' 2

e ^1,2, . . . , l + NoJ and

k t j 3 k Í j 5

( 1 4 ) 1 = \ .2 f o r j = l , 2 , . . . , n . From ( 1 4 ) we get

X'!1 = 1 for j=l , 2, . . . , n w h e r e m = k Jj J~ k <j J

and is a root of unity f o r j = l , 2 , . . . , n .

By t h e a s s u m p t i o n of our t h e o r e m if f o l l o w s that t h e r e exist s o m e j e -il, 2 , . . . , n s u c h that

FTTT 1

( 1 5 ) |Xj I > c ° From ( I S )

TTTT TTJT c

iri = max IX.I ^ c ° = e ° > 1 +

I^J S N J O

f o l l o w s and the p r o o f is c o m p l e t e .

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P R O O F OF T H E O R E M B. W e h a v e the w e l l - k n o w n i d e n t i t y C 1 Ő ) T r C A2) = T r2A - 2 Az .

On the o t h e r hand we h a v e

n 2 n '

C17> T r C A2) = X2 . X 2 = 2 [Re \.) - J (lm X . )

i = i

and

C I S ) Tr A

- ( X

t

x

n

) -

From C 1 6 ) - C 1 8 ) we o b t a i n

2

Re X.

T r C A2) = ^ T r2A +

C n — 1 ) T r A - 2nA.

t h u s

•I K - k *•••* k Y -

C n - 1 ) T r A - 2nA.

n

< 0.

S i n c e the left hand s i d e cannot b e n e g a t i v e for all real X^

then we get t h a t t h e r e e x i s t s at least i n e c h a r a c t e r i s t i c root which is a c o m p l e x n u m b e r .

From C 1 6 ) - C 1 8 ) w e o b t a i n

2

2 (Re A.)2- 2 (I- \ Y = k

i = i L = 1

2 Re X.

i = 1 C n — 1 > T r2A - 2nA,

and t h e r e f o r e we get

**2 ( i - x j * - 5 » J - K**

C 1 9 )

2

Re

X.

2 n A - C n - 1 ) T r A

2

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Let

C 2 0 ) d = max |Im X. |

l^iSn 1

T h e n from C 1 9 ) and C 2 0 ) we o b t a i n

n f 2 n A — C n — l ) T r2A

C21> k d2 £ 5 [lm J S - ,

i = i

w h e r e k^n if n is e v e n and k5n-l if n is odd. F r o m C215 we o b t a i n Cl.l).

Now, s u p p o s e t h a t C I . 2 ) is true and let

= (R e \Y~ (i m \ Y

Then we have

n n

C 2 2 ) J X2 = J xk = T r2A - 2 Az £ 0.

j =1 k = l

From C 2 2 ) we get t h a t t h e r e e x i s t s at least o n e c o m p l e x n u m b e r such that Im X ** 0. Let

n n ro

C23> 5 X2 = 5 B + 5 = T r2A - 2 A „

J k k 2 j = 1 k = 1 l=i

w h e r e B d e n o t e s the sum of s q u a r e s of all real e i g e n v a l u e s of A and let

min Ix I.

iSjSm { kjJ

I =i

C 2 4 ) x = man fx , x ,...,x 1 =

kd i^k.Sm I k1 2 mJ

J

Then from C 2 3 ) and C 2 4 ) we o b t a i n

C 2 5 ) m x, £ 5 x, 5 Tr A - 2A .

k . ^ k. 2

From C 2 5 ) we get

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( 2 6 ) xk d = [Re Xk J " - |im XkJ2 * L [Tr2A - 2 a J

w h e r e m^n if n is e v e n a n d míSn-1 if n is odd.

It. is easy t o s e e t h a t from ( 2 3 ) we g e t ( 1 . 3 ) and t h e r e f o r e the p r o o f o f T h e o r e m B is c o m p l e t e .

R E F E R E N C E S

[1] N. W. ICuharenko, On s p e c t r a l p r o p e r t y of a z e r o - t r a c e matrix, Mat. Z a m e t k i 3 5 (2), ( 1 9 0 4 ) , 1 4 9 - 1 5 1 (Russian).

[2 3 A. S c h i n z e l and H. Z a s s e n h a u s , A r e f i n e m e n t of t w o t h e o r e m s o f K r o n e c k e r , M i c h . M a t h . J . 12 ( 1 9 6 3 ) , 81-84.

[33 T. Szulc, On s o m e s p e c t r a l p r o p e r t i e s of m a t r i c e s with real e n t r i e s , Z . A n g e w . M a t h . M e c h . 68, ( 1 9 8 8 ) , 3 2 0 - 3 2 2 .

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