On a Determinant and Spectrum of some Matrix, I

(1)

ANNALES

U N I V E R S I T AT I S MARIAE C U R I E - S K Ł O D O W S K A LUBLIN—POLONIA

VOL. XXXIII, 19 SECTIO A ' 1979

Instytut Matematyki,UniwersytetAdamaMickiewicza,Poznań

4 . ■

Tomasz SZULC

nant and spectrum of the matrix:

A (1)

where А, В, C, D denote real, u

A

_2mx_2m

Oil a Determinant and Spectrum of some Matrix, I

О wyznaczniku i widmie pewnej macierzy, I Об определителе и спектре некоторой матрицы, I

This paper considers the problem of computing of determi-

B

* D.

>per (right) triangular

mxm matrices. It is well known ([1 ]) that the spectral radius condition ^(6)<1 (6 denotes a real mxm matrix) guaran

tees the convergence of the iterative process of the form (t) ^+1 “ Gxn» ^'a+'I’^n^® » n = G»^»***»

The oriterions for the convergence and divergehce of the process (2), generated by the matrix of the form (1J, are pre

sented below; it seems to us that these oriterions will be useful in some numerical calculations.

(2)

Let as mention that the process (2) can be also conside

red as an initial value problem for a first order ordinary difference equation ([4l). Oldenburger has found ([5]) the connection between the spectral radius of G and the stabill ty of the trivial solution of the equation (2). Some corolla

ries for stability of the trivial solution of the equation (2) are also presented in this paper.

LEMMA 1» Matrix of the form (1) is reducible.

In order to prove this fact it is sufficient - taking into aooount the theorem given in [5] (p. 50) - to point out such index subset JC [1,2,...,2m } that

<5> ^2mx2nk»3l for any pair (1,J), (14 J, J) . It is easily seen that for the set J = {m,2m} the*

oondition (5) is satisfied. So, Lemma 1 is proved.

THEOREM 1. for the matrix (1) the following formula

holds

det *^2mx2m “ П det i»1

ai,i bi,i

ci,i di,i

Proof by induction. Рог а в 2 we havet

» - >

A

_4X4

*1,1 a1,2 0 ®2,2

'1,1 0

'1,2

’1,1 0

“2,2

®1,2 <4,1 *1,2 '2,2 "2.2

(3)

On a Determinant and Spectrum of Some Matrix, I 235 Since matrix

formation P has the form

(4)

where B„

^4x4 reducible then there exists a trans- (by "orthogonality") such that matrix P^k4PT

B,

B_1,1

©

B.1,2 B2,2

■*11’ a2 2 denote square matrices and & -nullmatrix.

Let us consider the matrix P = [e^,ej,e2,e^], where e^

(i= 1(1)4) denote unity vectors of real 4—dimensional space

r\ Matrix P is non-singular and has the inverse of the form

,-1

eT®3

T

Since the transformation by "orthogonality" does not change the value of determinant we have

deti/l^*^ s det PA^.^ =

= dot

• r- «

10 0 0 a1,1 *1,2 b1,1 b1,2 10 0 0

0 0 10 0 a2j2 0 b2,2 0 0 10

0 10 0 °1,1 °1,2 *1,1 d1,2 I 0 10 0 0 0 0 1 0 °2,2 0 *2,2_ ^{0 0 0 1}

a1,1 f 2 ,1 12 - det '1,1

0

’1,2 a2,2 s2,2

*1,1 0

Q1,2 b2,2

"2.2

0 0 1 0

0 1 0 0 0

(4)

det

*1,1 b1,1 a1,2 b1,2

°1,1 *1,1 °1,2 d1,2 O

O

*2,2 °2,2

°2,2 <*2,2 Hence, by [-2] (p. 52) we obtain:

det x 4 . det

- -

*1,1 b1,1 *2,2 42,2 det

®1,1 ^,1 c2,2 *2,2 2

Hf det

1.1

ci,l di,i

So, Theorem 1 for m . 2 ls proved.

Let U8 now assume that

(5) det Ag m *2m j~| det

ai,i bi,i 1=1 ci,i di,l we are going now to show that

m+1 d9t ^2(m+1) x 2(m+1) ' FI defc

1=1

al,i bi,l

where

A2(

^d

+1)

x2(m+1) =

ci,i di,i

dm+1,m+1

(5)

On a Determinant and. Spectrum of Some Matrix, I 237

r^J

By Lemma 1, there exists a transformation P (by "orthogona

lity") such that x 2(m+1)^T has the form Xet us consider the matrix P = [®i’®2’* * * ’®m+2’* * *

’»*’ e2m-1’e2m’enH-1 ,q2iih-2]’ where ei = K1)2m + 2) are unity vectors of real 2m+2-dimensional space ROnu?

sP =a/

80«

Aw» x /MQt

P'H2(m+1) x2(DH-1)r 1,m a1,55T

m,m m,m+1 c1,1 c1,m-1 c1,m C1,m+T

dm,m a*,m+1 d1,1“,m=T ,~m~ d1 ,m+1

m-1 ,m+1 dm-1,m-1 dm-1 ,m ^m-l ,m+1

m+1,m+1 0 0 »-1,-1

m+1 ,m+1 0 dm+1 ,o+1

0

(6)

®1,1 a1,m^T~a1,m b1,m b1,1 b1,m^I~a1,m+1 b1,c+1

O *m-1,m-1 aiu-1,m °m-1,m ^bm-1,m-1 am-1,m+1 bm-1,m+1 O

O

a,m,n m,m m,m

O .m O

am,m+1 Dm,m+1

°m,m+1 dm,m+1

cm-1,m-1 cm-1,m dm-1,m cm-1,m+1 ^ra-1,ci+1

O

o o

am+1, m+1 bm+1, m+1 c m+1, mt-1 dm+1, m+1

or in a much closer form

(6)

PA2(

bh

.1)

x2(m+1)P =

where R2 x2 -

2m x 2m V®2 x2m am+1, m+1 bm+1, m+1 CIfl+1 ,EH-1 ^m+l.ffi+l, the form (4). Tlius, by [2] (p. 52) we get

(7) det PxA2fm+.tjx2(nH.1)PT = <łet4mx2mdet

A2m x 2

'2X2

{ so, the matrix (6) has

am+1,m+1 bm+1,mł-1 .cm+1. m+1 dm+1. m+1

(7)

On a Determinant and spectrum of Some Matrix, I 233 Evidently:

det P^2(iw.1)x2(bh-1)^T “ det ^2(w1)< 2(m+1) • so with respect to (7) we obtain

(8) det'A2(nH-1)x 2(bh-1) “ detAzax2m det

am+1, m+1 ^m+1 , m+1 cm+1 ,hh-1 dn+1 ,«H-1 Let us now use the transformation P (by "orthogonality"),

r» 9C 5S> ft, » » -l W

P s Le1,e2,,,.,em,em+2,emtj,...,e2ni,e]n+1j, where e^

(is 1(1)2m) are unity vectors of real 2m-dimensional space

(8)

Honce

, « «T A

det PA>mx2m p = d<3t X 2u

(9)

On a Determinant and Spectrum of Some Matrix, I 241 but

so (9)

det P A2o x 2a PT = det ^2m x2m ,

det ^2m x 2m a det ^2m x 2m * Taking into account (5), (8) and (9) we get

det ^2(m+1) x 2(m+1) = det ^2m x2m det

am+1, m+1 bm+1, m+1 Cm+1,m+1 dm+1,m+1 am+1, m+1 bm+1, m+1

cm+1, m+1 dm+1, m+1 and the proof of our Theorem is completed.

« det A>m x2m det

m+1

PI det 1=1

a i,i bi,i

°1,1 di,i

THEOREM 2. The spectrum of the matrix (1) is equal to tho union of the spectra of the matrices of the form

ai,i • bi,l

ci,i di,i

1 = 1(1)m,

k2m < 2m} = U 6 ( al,l bi,i 1=1 ®i,l di,l Because the eigenvalues of the• l.e. )•

42m x 2m are roots of the equation

det[4mx2m' Al2mx2m ] = 0

and matrix ^2mx2m- Al2mx2m has the form (1), so, by previous Theorem, we have

i

(10)

m det [^2mx2m " ^^mxRml = TH det

’ ia1

m s det

ia1

ai,l ci,l

°i,i di,i

ai,i bi,i L°i,i di,i That proves the Theorem.

From Theorem 2 and Kakeya. theorem ([6]) it follows:

COROLLARY 1. If there exists such number 1

that

O<1 <- + dj- jXdet

al,l bl,l

°1,1 ^,1

then matrix vAgm x 2m 1x38 an O^-Genva^ue with modulus greater than one.

COROLLARY 2. If for any 1 (la 1(1)m) the inequalities

1 > - (Sj -^ + d1^1)>det

al,l bl,l lcl,l dl,l

>0

” I2 x 2 )

are satisfied, then the spectral radius of 42m x 2n is le3S than one.

REMARK 1. Corollary 1 gives a simple criterion for the divergence of the Iterative proces (2) generated by the matrix

^2mx2m *

By Oldenburger's theorem and the form of cpectrum of

^2m x 2m we i-^ßäiately obtain

(11)

On e Determinant and Spectrum of Some Matrix, I ₂₄₃ THEOREM 3. The trivial solution of difference equation (2) (with matrix '^'2mx2m^ asymptotically stable iff for rny i ,(i = 1(1)m) all roots of equations

(10) \ *- (a^ 1 î + ®i iî i ” **i iî i =

have moduli less than one.

Prom Paddiejewa's theorem (LlJ) pp. 100-101) and from the form of spectrum of x-2m we ot,^'a^n,

I THEOREM 4. Iterative process (2), with matrix ^-2mx2m’

is convergont iff for any i (i = 1(1)m) all roots of equa

tions (10) have moduli less than one.

REJJIARK 2. Because the transformation by "orthogonality"

does not change the value of determinant and the spectrum of matrix, thus Theorems 1, 2, 3, 4 and Corollaries 1, 2 are va-

where M, N, P, Q denote lower (left) and A, B, C, D upper (right) real, triangular mxm matrices and R,S-real,' mxm matrices of the formst

(12)

respectively; the transformation P (by "orthogonality”) has the form-

(11*)

(12')

(15')

(14')

(1$')

P 0

mxm (JT ,T

m xm in x m

mxm

in x m Pm xm

a xm

Pm xm ®m xm

mxm •nT

m xm mxm0.

(p®

(p®„

©,in * m

mxm

mxm 0_mxm

0,mxm

mxm mxQr m )•

).

0

p 0

0

0 0 m x m m x m

0m x m ).

m x m 0mxm

for the matrices: (11), (12), (15), (14) and (15)» respective

ly (P = [em,... ,e1 ] and ®n xU is the nullmatriz).

For example:

P(1l')^'2mx2mP(11')T = 0 mxm’

Blxii

0._ax_hi

0_nxn ^»»LI N

*■—

p Q

k 1

A B C D

(13)

On a Determinant and Spectrum of Some Matrix, I ₂₄₅

REFERENCES

Cl] Faddiejowa,W.N., Metody numoryczne algebry liniowej, Warszawa 1955.

[2] Lancaster, P., Theory of matrices, New Tork - London 1969.

[3] Oldenburger, R., Infinite powers of matrices and characte

ristic roots, Duke Math. J., 1940, 357-561.

[4] Ortega, J.M., Stability of difference equations and conver

gence of iterative processes, SIAM J. Numer. Anal., 10(2) (1975), 268-282.

[5] Ortega, J.M., Rheinboldt, W.C., Iterative solution of non

linear equations in several variables, New Tork - London 1970.

[6] Turowioz, A., Geometria zer wielomianów, Warszawa 1967.

STRESZCZENIE

W pracy podano nowe kryteria zbieżności względnie rozbież

ności procesu iteracyjnego postaci Хд+1 = Gxn, generowanego przez macierz G.

Резюме

В работе представлены новые критерия сходимости или рас

ходимости итерационного процесса типа хп+7 =Gxn; порождаемо

го матрицей G .

(14)

Nakład 650+25 egz. Ark. wyd. 12, ark. druk. 15,5. Papier offset, kl. V, 80 g. Oddano do powielenia w lutym 1981 r., powielono w kwietniu 1981 r. Cena zł 38,—

Tłoczono w Zakładzie Poligrafii UMCS w Lublinie, zam. 100/81, W-3

*

(15)

ANNALES

UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN — POLONIA

VOL. XXXI SECTIO A 1977

11. Z. Lewandowski, R. Libera, E. Złotkiewicz: Values Assumed by Geller Functions.

O zbiorze wartości funkcji Gelfera.

12. K. W. Morris, D. Szynal: On the Limiting Behaviour of Some Functions of the Average of Independent Random Variables.

O granicznym zachowaniu się pewnych funkcji średniej niezależnych zmiennych losowych.

13. Z. Radziszewski: On a Certain Interpretation of Linear Connection on a Differentiable Manifold.

O pewnej interpretacji koneksji liniowej na rozmaitości różniczkowej M.

14. R. Smarzewski: On Characterization of Chebyshev Optimal Starting and Transformed Approximations by Families Having a Degree.

Twierdzenie charakteryzacyjne dla optymalnych aproksymacji starto

wych i transformatowych elementami rodzin nieliniowych.

15. J. Stankiewicz, Z. Stankiewicz: Some Remarks on Subordination and Majorization of Functions.

Pewne uwagi o podporządkowaniu i majoryzacji funkcji.

16. J. Stankiewicz: Quasisubordination and quasimajorization of Analytic Functions.

Quasipodporządkowanie i quasimajoryzacja funkcji analitycznych.

(16)

ANN ALEJ UNIVERSITATIS MARIAE C LUBLIN — POI

VOL. XXXII SECTIO A

w Lublinie

W50I33

CZASOPISMA

■ 1. F. Bogowski, Cz. Bucka: Sur une cl

O pewnej klasie funkcji gwiaździsty

2. P. Borówko: On the Stability of Solutions w* mutn.».,*«. u4uu»v.m

Random Retarded Argument.

O stabilności rozwiązań równań różniczkowych z losowo opóźnionym argumentem.

3. T. Havarneanu: On a Certain Integrodifferential System with Delay.

O pewnym układzie równań różniczkowo-całkowych z opóźnieniem.

4. J. Janowski, J. Stankiewicz: A Relative Growth of Modulus of De

rivatives for Majorized Functions.

O względnym wzroście modułów pochodnych dla funkcji zmajoryzo- wanych.

5. J. Kaptur: On Certain Boundary Value Problems for Partial Differential Equations.

O pewnych problemach brzegowych dla równań różniczkowych cząstko

wych.

6. T. Kuczu mow: An Almost Convergence and its Applications.

O prawie zbieżności i jej zastosowaniach.

7. I. D. Patel: Recurrence Relations for Moments of Inflated Modified Power Series Distribution.

Wzory rekurencyjne na momenty dla zmodyfikowanego rozkładu sze- regowo-potęgowego typu inflated.

8. M. Polak: On Properties of Some Classes of Discrete Distributions.

O własnościach pewnych klas rozkładów dyskretnych.

9. W. Rom pa la: Liftings of «-Conjugate Connections.

Podniesienie koneksji «-sprzężonych.

10. M. Turinici: Sequentially Iterative Processes and Applications to Volterra Functional Equations.

Ciągowe procesy iteracyjne i ich zastosowanie do równań funkcyjnych Volterry.

11. M. Wesołowska: Some Aspects of the Theory of Experimental Designs.

Pewne aspekty teorii układów eksperymentalnych.

12. J. Zając: The Ahlfors Class N and Its Connection with Teichmuller Quasi- conformal Mappings of an Annulus.

Klasa N Ahliforsa i jej związek z odwzorowaniami quasikonforemnymi Teichmiillera w pierścieniu.

A d r e s s e:

UNIWERSYTET MARII CURIE-SKŁODOWSKIEJ BIURO WYDAWNICTW

Plac Marii

Curie-Skłodowsklej 5 20 031 LUBLIN POLOGNE

Cena z! 36,