NATIONAL BUREAU OF STANDARDS REPORT

1162

BIBLIOGRAPHY

ON

BOUNDS FGB CHARACTERISTIC ROOTS

OF FINITE MATRICES

September 1951 Olga Taussky

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

E. U. Condon, Director

THE NATIONAL 8UREAU OF STANDARDS

The scope of activities of the National Bureau of Standards is suggested in the following listing of the divisions and sections engaged in technical work. In general, each section is en- gaged in specialized research, development, and engineering in the field indicated by its

title. A brief description of the activities, and of the resultant reports and publications,

appears on the inside of the back cover of this report.

I. ELECTRICITY. Resistance Measurements. Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Electrochemistry.

2- OPTICS AND METROLOGY. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Gage.

3* HEAT AND POWER. Temperature Measurements. Thermodynamics. Cryogenics. Engines and

Lubrication. Engine lAiels. Combustion.

t. ATOMIC AND RADIATION PHYSICS. Spectroscopy. Radiometry. Mass Spectrometry. Physical Electronics. Electron Physics. Atomic Physics. Neutron Measurements. Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Instruments Branch.

5. CHEMISTRY. Organic Coatings. Surrace Chemistry. Organic Chemistry. Analytical Chemistry. Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

G. MECHANICS. Sound. Mechanical Instruments. Aerodynamics. Engineering Mechanics. Hydraulics. Mass. Capacity, Density, and Fluid Meters.

7. ORGANIC AND FIBROUS MATERIALS. Rubber. Textiles. Paper. Leather. Testing and

Specifications. Organic Plastics. Dental Research.

8. METALLURGY. Thermal Metallurgy. Chemical Metallurgy. Mechani cal Metallurgy . Corrosion.

9. MINERAL PRODUCTS. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Building Stone. Concreting Materials. Constitution and Microstructure. Chemistry of Mineral

Products.

10. BUILDING TECHNOLOGY. Structural Engineering. Fire Protection. Heating and Air Conditioning. Exterior and Interior Coverings. Codes and Specifications.

11. APPLIED MATHEMATICS. Numerical Analysis. Computation. Statistical Engineering. Ma- chine Development.

12. ELECTRONICS. Engineering Electronics. Electron T\ibes. Electronic Computers.

13. ORDNANCE DEVELOPMENT. Mechanical Research and Development. Electromechanical

Fuzes. Technical Services. Missile Fuzing Research. Missile Fuzing Development. Projectile Fuzes. Ordnance Components. Ordnance Tests. Ordnance Research.

I

1!. RADIO PROPAGATION. Upper Atmosphere Research. Ionospheric Research. Regular Propagation Services. Frequency Utilization Research. Tropospheric Propagation Research. High

Frequency Standards. Microwave Standards.

15. MISSILE DEVELOPMENT. Missile Engineering. Missile Dynamics. Missile Intelligence.

Missile Instrumentation. Technical Services.
NBS PROJECT NBS REPORT

9011 1162

BIBLIOGRAPHY

'ON

BOUNDS FOR CHARACTERISTIC ROOTS

OF FINITE MATRICES

September 1951 Olga Taussky

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

This report is issued for

in sny form, either in whol

frost the Office of the Dire

reprinting, or reproduction

sion in writing is obtsined

ngton 25, D. C.
• .
This is a revised edition of part of an earlier

bibliography prepared in 1949 by A« Ostrovski,

0* Taussky Todd, <J 0 Todd which included section's on

Inversion of Matrices and computation of characteristic KootSe These have not been revised in view

of the fact that more complete bibliographies are

in preparation o In particular we refer to the contribution by G* ha Torsytbe to the Proceedings of

the Symposium on Simultaneous Linear Equations and

the Determination of Ligenvalues « {National bureau

of Standards Semi~centenial 1951)

-1 -
Bounds for characteristic roots of matrices

are of importance from various aspects 0 For instance in some methods of computing the characteristic

roots as approximate value is required t a reasonable value may be found if the bounds are known 0 On the other hand, it is frequently necessary, e,go

in stability problems, to know whether the roots of

a matrix are inside the unit circle or whether they

have positive or negative real or imaginary parts.

A number of results in this connection has been obtained. Some of these results give bounds for the

absolute values of the roots, others deal with

regions in the complex plane inside which the roots

must lie. in some cases the regions or bounds are

invariant under unitary transformations. An

important classical result which comes under the

subject "Bounds for characteristic roots of

matrices" is Frobenius theorems a matrix with

positive elements has a real and positive

number as dominant characteristic root.

a

- 2 -
S. £1, Afriat, Bounds for the characteristic values

of matrix functions, Oxford quart# tJ« &, 81

to 84 (1351)

S. £1 , Afriat, An iterative process for the numerical

determination of characteristic values of

certain matrices, Oxford quarterly J. 6 , 121 to 122 11951)

P, Alexandroff and n. nopf, Topologie, Springer

1935, p, 480 o

B, W. Barankin, Bounds for the characteristic roots

of a matrix. Bull. Amer. Math, Soc 51, 767 to

770 (1945)

V* Bargmann, 0, Montgomery, J, von jueumann, Solution

of linear systems of high order, Institute

for Advanced Study, Brinceton, £l, J., 1946

«f* Bendixson, Bur les racines d'une equation fon-

* damentale, Acta Math, 25, 359 to 365 (1902)

A# Brauer, Limits for the characteristic roots of

a matrix, Luke Math, «Jour. 13, 387 to 395

(1946)

At Brauer, Limits for the characteristic roots of

a matrix II, Luke Math, Jour, 14, 21 to 26

(1947)

A« Brauer, Limits for the characteristic roots of

a matrix ill, Luke Math, Jour, 15 871 -to 877

(1948)

A* Brauer, Limits for the characteristic roots of

a matrix IV, applications to stochastic

matrices, to appear in Luke Math, jout»

Tt J $ 1

7 Jk Bromwich, on the roots of the characteristic equation of a linear substitution, Acta

' Math, 30, 297 to 304 (1906)

?9 Browne, xhe characteristic equatioh of a

matrix. Bull, Amer, Math, boo, 34, 363 to 368

(1928)

1, I', Browne, i‘ne characteristic roots of a matrix,

Bull. Amer. Math, Boo, 36, 7 05 to 710 (1930)

g, T§ Browne, Limits to the character is-cic roots

©f a matrix, Amer 0 Math, Monthly 4o> 252 to

, 265 11939

)
B. L o uollatz, ifiinschlies sungs satz fur die charahteristischen zahlen von Matrizen, Math®

Zeitschrift 48 , 221 to 226 (1942)

Ha uourant and d® Hilbert, Methoden der mathematj sehen nhysik, ad. i, 26, Springer, 1931

Uc Uc uurrre, uassrni ovals associated with a

second order matrix, Amer „ Math® Monthly

5J5, 487 to 489 (1948)

II. Dmitriev and Do Dynkin, On the characteristic

roots of a stochastic matrix*, UoKo Acadl

Sci . UoKoSoSo (NeSe ) 49 3 159 to .162 (1945)©

N. Dmitriev and Do Dynkin, On the characteristic

roots of stochastic matrices, bulletin

Acado ocio UcKc bicbo^ sere Matho 10, 167 to

184 (1946)

Ko Dan, On a theorem of weyl concerning eigenvalues

of linear transformations I, Proc 0 flat 0 Acad.

c: cio USA 35, 652 to 655 (1949).

II, iroco JMat. Acado Sci. USA 36, 31 = 35 (1950)

Problems 4429 and 4430 in American Math.

Monthly 58, 194 to 195 (1951).

A. a* Darnell, Limits for the characteristic roots

of a matrix, Hullo Amer® Math. Soe® j50 , 789 to 794 (1944) 0

, h. Darnell, Limits for the field of values of a

matrix, Amer. Matho Monthly 52, 488 to 493

i 1945 ) #i L Fischer, .. Uber quadratische Dorraen mit reeXXen

Koefflzi eaten, Monatshefte f® Matho u. Phys*

,16, 234 to 249 ( 3905 ) * K- Frechet, Oomportement asymptotique des solutions

i un system© d equations lineaires et

homogenes differences finies du premier t ordre a* coefficients constants, Publo J'acuite

/Sci- UniVe Masaryk^l.78, 1 to 24 (1933)

M® Frechet, Hecherches theoretiques modernes sur la

theorie des probabilites 2, Paris 1938 0
Go Probenius, Uber Matrizen aus positiven Element en9 Berliner Sitzungberichte*, 471 to 47 S (19C8)

Go Fxobenius, Uber Matrizen aua poaltiyen. Blementen* H9 Berliner Sit zungbe r i cb f e , 514 to 518 (1S09)

Go Frobenius, Uber Matrizen aus nicht negativen

Elementen, Berliner Sitzungoerichte, 456 to

477 (1912)

Fo Gantmakher and Mo Krein, Sur ies matrices completement non negatives et oscillatoires,

v

ComposjLtio Math* 4, 445 to 476 c(1937)

So Gersgorin* Uber die Abgrenzung der Eigenwerte

einer Matrix, 9 Izv 0 Akad* I\lauk SoSoSoR

VII , 749 to 754 (1931)

Givens* Fields of values of a matrix* to appear

in Proco Amer« Math* 3oc 0

F« Hausdcrff* Der Wertevorrat einer Bilinearform,

Matbo Seitschrift j3, 314 to 316 (1919)

A® Hirsch, Sur les racines d’une equation fondamentale, Acta Math* 25*, 367 to 370 (19021

%

Kojima, Tetsuzo, On a theorem of Hadamard v s and

its application*, Tohoku Math® J our e 5^54 to

60 (1914)

Wo Ledermann, On a problem concerning matrices with

variable diagonal elements s Proc & Royal Soc G Edinburgh 60 , 1 to 17 |1940J'

Wo Ledermann, On an upper limit for tne latent

roots of a certain class of matrices*, Jour 0 London Math* Soc e , 12 9 12 to 18 (193?)

.Wo Ledermann, Some mathematical remarks concerning

boundary conditions in the factorial analysis

of ability, Psychometrika 1 9 165 to 174

(1936)

W« Ledermann, On the asymptotic probability distribution. for certain Markoff processes,

Proc» Canto Philo Soc 0 46 , 581 to 594

(1950)

-5
W® Ledermann, Bounds for the greatest latent roots

of a positive matrix, J* London Math® Soc«

25, 265-268 (1950)

W® Leaermann, Asymptotic formulae relating to the

physical theory of crystals, Proc 0 Royal

I3oc* London 182 9 362 to 377 (1944);

Fo Do Mur^aghn, On the field of values of a square

matrix, Proc 0 Hat • Acado 18, 246 to 248

(1932)

Ro Oldenburger, Infinite powers of matrices and

characteristic roots, Duke Math® «J'our® _6,

357 to 361 (1940)

A® Os trow ski, wUeber die Determinanten mit uber=

wiegender Mauptd i agonal e , Comm 0 Math®

helv«, 10 , 69 to 96 (1937)

Ac Ostrowski, "Sur la determination des bornes

inferieures pour un.e classe de determinants”

Bull*, des Be® Math., 61 , 1 to 32 (1937)

A s Ostrowski, uber das Nichtverschwinden einer

Klasse von. Determinanten und die Lokalisierung oer charakteristiscjien Wurzeln von

Matrizen, to appear in Comp® Matho

A * ( strowski, Sur les conditions generales pour la

regular it e des matrices, to appear in Proc Q of Sever! convention, Kome, 1950©

Ac Ostrowski, Sur quelques applications des fonctions

convexes et concaves au sense de I® schur,

to appear in Montel jubilee volume 1952

Wo

Wo

Wo

Wo

Vo Parker, The characteristic roots of a rmn :< Duke Math® Jour© 3, 484 to 487 (19 37)

V 0 Parker, ^Limits of the characteristic roots

of a matrix”, Duke Math 0 Jour®, 10, 479 to

482 (1943)

V* Parker, The characteristic roots of a matrix

Duke Math® <7our 0 , 12. 519 to 526 (1945)

Vo Parker, The characteristic roots and the

field of values of a matrix, Duke Matho

Jour,, 15, 439 to 442 (1948)

Vo Parker, characteristic r >ots and fields of

value of a matrix, Bull® Arner© Math® joc

57, 103-108 { 1951)

6 -
M* Parodi, vSur uiie application d

! un theoreme de

M» J* Radumara, hull. Sci,, Math- (2) 72

136 to 138 11948, ^ x M« Parodi, Application uAun theoreme de M. hadunard

a 1* etude de la stabilite' dec systenes, 0©

A » Acad. Sci® Paris 228 , 807 to 808 (1949)

M* Parodi, Sur la determination d’une limite

superieure de la par tie reelle des racines

de 1 * equation aui frequences propres d*un

reseau electrique, Go R 0 Acad© Sci© Paris

228 , 1400 to 1402 (1949) , M. Parodip Remarque aur la stabilite , C© R 0 Acad©

Sci. Paris 228 , 51 to 52 (1949) y

M. Parodi, Complement d un travail sur la stabilite

,

C. R e Acad® Scio Paris 228 , 1198 to 1200 (1949)

M* Parodi, Sur une application d’un theoreme de r Me Muller, Bull. Sci 0 Math® (2) 73, 192 to

196 (1949) , , M. Parodi, contrioution a l

ff etude de la stabilite,

i*'.i « h Pays® Radium 18) 10 , Parodi, Sur une propriety d*

appl i c a t i on a

1 .I

s dVud e d

348, to 352 (1949)

un equation algeorique;

gs oscillations dans

H.

Me

M.

Me

eti ux a 1 e c tr i nie s , C R , A c A9 , 1190 to 1192 (1949)

J.O O -La

les reseaux alectri >ues» C R, Acad

Aaris

Parodi, Sur

des e qua t

i

ob=b~ al gebri q ue s, Bui IvSci# Ma th

(2) 73, 135 to* 14^ 1/949)' , Parodi, sur une propriete des determinants gauches

uhLl-es des modules des r a cine A

Ann. SoCe 3c i e Bruxelles, Rer 1. 63 81 to 82

(1949) ,,

Parodi, Quelques proprietes jies matrices H, Ann c boc , Scio Bruxelles, Ser 1„ 64, 22 to 25

( 1950 ) . / , Parodi, Sur, la- definition des reseaux electriques

mailles dont Inequation aux frequences

propres a

v ses racines a 1

s inter ieur d 9 un

circle doiine, 7« Phys 0 Radium (8) 11 , 141

to 143 (I960)

Parodi, Sur "nine limite superieure du rapport

des valeurs car&cteris tiques de aeux

matrices symetriques, d/finies positives,

a

v elements reels, dont les elements

correspondents different peu u. K, Acad*

Sci. Paris 230, 705 to 7C7 (1950)

9
0, Perron, Zur Theorie der Matrizen, Matho Ann alen

64, 248 to 263 J1907)

Go Perron, Grundlagen fur eine Theorie aes

J a k

o

d isohen Ke 1 1 e nbruc halgs or i thmu s , Math

©

Annalen 64 1 to 7 6 (1907)

G© Pick, uber die Wurzeln der charakteristischen

Gleichun&en von Schwingunasproblemen, ZAIM

2 353 to~357 \ 1922

)

V 9 k

O

manov sky, bur les zeros des matrices stocastique,

U e K s Acad. Scic Paris 192 , 266 to 269 *1931)

Hans Schneider, An inequality for latent roots

applied to determinants with a dominant

diagonal, manuscript©

1© Schur, uber die charakteristischen Wurzeln einer

linearen suostitution mit einer iuiwendung

auf die Theorie der mtegralgleichungen,

Mathematische annalen 66_, 488 to 510 ^1909)

Wo specht, Zur Theorie der algebraischen Gleichungen,

dahresber do Deutsche matho vereinigg £8

142 to 145 i 1938)

p 0 Stein and K© L. Kosenberg, un the solution of

linear simultaneous equations by iteration,

A London Math® boc. 2^3, 111 to 118 ^1948)

Pe btein, a note on the bound of the real parts of

the characteristic roots of a matrix,

manuscript © P © Stein, A note on bounds for multiple characteristic

roots of matrices, manuscript v

Z# szmydtowna, bur les racines caracteristiques et

sur les directions caracteristiques de

certains matrices, Ann e boc, Pol© Math© 22

235 to 240 { 1950)

0© xaussky, Bounds for characteristic roots of

matrices, uuke math* oour. 15, 1043 to 1044

V 1948 ) 0 * xaussky, sounds for characteristic roots of matrices

al, do of research, i\ibb 46, 124 to 125 *1951;

-8
u. Taus sky3 a method, for obtaining bounds for characteristic roots of matrices with application

to flutter calculations* Aeronautical

Kesearch council, or eat .Britain, report

10508, (1947)

0# Taussky, a recurring theorem on determinants,

Amer. Math* monthly 56, 672 to 676 (1949)

0. Toeplitz, uas algebraische Analogon zu einem

batze von jrejer, Math«Z 0 2^187 to 197 (1918)

A. U* walker and j 0 D c Weston* Inclusion theorems

for the eigenvalues of a normal matrix,

dour. London Math, boc. 24, 28 to 31 (1949)

h* weyl, inequalities between the two kinds of

eigenvalues of a linear transformation,

rroc. JNat. Acad. Sci, 35, 408 to 411 (1949)

h* wielandt, Lin Linschliessungssatz fur charakteristische wurzeln normaler Matrizen p Arohiv.

do Math. 1 , 348 to 352 (1949J

h* Wielandt, Lie jiinsohliessung von Ligenwerten

normaler Matrizan, Math© Annalen 121, 234

to 241 (1949)

h. wielandt, unzerlegbare, nicht negative matriazen

Math. Lu 52, 642 to 648 (1950)

H. wielandt, z»ur Abgronzung der sslbst adjunglerten

aigenwertaui^aben Aaume ©ndliehar Dimension* Math. jNachriohten 2, 328 to 339

U949)

h. wittmeyer, jsiingluss dor And e rung elner Matrix

auf die LQB\m§ das zugthorigen uloiohungs*

systems, sowi© auf di© oharakt©riitisoh@n

xahlen und di© Mgenvektoren, ZAMM 16, 28?

to 3QQ f} (193i) 6 hi wittm©y©r P ub©r di© mosung von lisatuna

01©iehun§i§yst©m@a duroh Iteration* ZAMM

i|, 301 to 310 (1136)

- 9 ®
Addenda:

A, brauer, bounds for characteristic roots of

matrices, to appear in Proceedings of the

symposium on Simultaneous Linear liquations

and the determination of bigen values*

A* Ostrovski, bounds for the greatest latens root

of a positive matrix, manuscript*

ti 9 wielandt, Inclusion theorems for eigen values,

to appear in proceedings of the Symposium

on Simultaneous Linear liquations and the

Determination of hiigen values

10-
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The National Bureau of Standards is the principal agency of

the Federal Government for fundamental and applxed phys-

( the MRS work is performed for other government agencies,

portion of the NBS work is periorm

Energy

particularly the ^.u o£ Standards are set

n li t cio iQth The scope of activities is suggested in

s .S «“Ln, . «h. i»i* -f «». front

cover.

Reports and Publications

The results of the Bureau’ s work take the form of either actual

’ssuedTo the Spring ^ot

Published papers a PP^either in ^

scientific socle ties.

E^LHitP^t#SOmce:

Technical N... Wl.ti.. potions,

radio communications throt^hout t e worl

d ,,,,01101101 Seports and Miscellaneous Publications.

Information on the Bureau

1 , publication., can be found in WS

a£3 s^62:

of Standards, Washington 25, D. C.