NATIONAL BUREAU OF STANDARDS REPORT

1964

ON NEARLY TRIANGULAR MATRICES

By

A, M. Oatrowsfci

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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@ Office of Basic Instrumentation

»

x

t

• Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1102-20-1104 October 1,1952 1964

ON NEARLY TRIANGULAR MATRICES

by

A. M. Ostrovski

American University

University of Basel

The publication, reprlnl

unless permission is obti 25,D,C, Such permlss

cal ly prepared If that t

Approved for public release by the ^ „ [ralwllj

Director of the National Institute of „d ,, w„lln81 „ n Standards and Technology (NIST) has been speciti- on October 9, 2015 t for its own use,
: «

f
On Nearly Triangular Matrices*

by

A. M. Ostrovski

American University and University of Basle, Switzerland

Introduction

Consider a system of linear equations

n

( 1 ) ^ a x - X ^

(•* * t*

( u

~~

j i

with the matrix A, in which all diagonal elements

^ 0 and the elements off the. diagonal satisfy for

m, M tlie inequalities

a^ ( ff = 1, . .

. ,n) are

two positive numbers

If n is very small, the system does not essentially differ from the

corresponding' "triangular" system in which all a^ with V<^ are re

placed by zeros and the matrix of which will be denoted by A^ ^

. It

appears then plausible that the solutions of this "triangular" system

does not differ very much from that of the system (1)

However the value of the determinant of the order n
1 -M 0 0

0 1 -M

\

X

X

X

0

'Vr. O

X

X

X >* ^ 1

0 1

- 1-mMVI-1

shows that if M is fop instance &10, the determinant of our system vill

^ n -(n-l) a not be even necessarily different from zero unless m< 10

detailed study of the problems connected with the matrices characterized. by (2) appears therefore to be of importance and. interest.

As the first problenTbln this connection we give a necessary and suf

ficient condition for the determinants of all matrices A satisfying

the conditions (2) being non singular. This condition is, if m < M,

given by

( 3 ) m <

M

( 1 + m)n

(

' i

and, if m = M, by

( 3 °) m <C n-l

In order to obtain a precise measure of the influence of the change

from A to a(°), we have to discuss the estimates for convenient norms

of the matrix A

^ - A^^

i-je consider in particular two such norms defined in the section IV

and denoted b;

a -1

>u

y

( °) Ip (p = 1 , 00 ), which are particularly

i table for the problems of Numerical Analysis. Ve show then

} In

assuming witho.ut loss of generality that ^uu.

~ ^ (.
that for given values of m, M we have, if M ^

1,5 n£4-:

n

(4) | A'

1- ^I = (l + M)" jfj

nTM t

§ (M£

where 1- Sis the smallest modulus of the determinant attainable for

the matrices A and is connected with m by the relation

(5)

Mn m

= 1-0 m > 0 <0 <1

where

( 6 ) Mn

(+M) n

-n M-l

M

If M K —•

_2_ . the formula (4) need not be valid any longer, but we

n

can prove In this case the relation

(7)

valid as long as m remains less than

2n

<i.f

The estimate (7) is not a "best" estimate for all values of Mi n but still it is not very far from the best, since for m = M < J * — n-1

we have

I -l £ . ./n-l'im..( m = n-i) ?

I ft - A l„ ^ l-Cn-lIm ^

( 8 )
which cannot be improved for any value of m , The condition ( 3 ) is derived in the section III of this paper

y theorem B. However, we derive it as a special case of a more general

theorem concerning the case where in the inequalities (2) the expressions m, M depend on ^

, that is to say, change from one row to

another. The necessary and sufficient condition of all matrices A

being regular ( theorem A, section III) is in this case ether unwieldly,

but may be still very useful in some cases, since it contains 2n-2

instead of two essential parameters. The direct derivation of theorem

B is of course much simpler since, as the reader will immediately see,

the computations of the determinant in the section I can be considerable shortened in this case. The connection between the formal

algebra of the section I and II and the theorems A and B is provided

by a result concerning the so-called H -determinants and M -determinants

&

which I published ]g years agd[l]. The results about the norms

I fdO—l I

I r\ ~

Ip are obtained in using the explicit representation of

the inverse matrix of a certain matrix A* which provides a majorant

1 —1

for all matrices A". The formulae giving0 are derived in the

n

\ -1 l"1- | second part of the section II , and in the section V the norms j ^ «i -1^ n ‘p

are derived and discussed. The corresponding inequalities for

1 A-i .Co>l\

I n “A Ip are then obtained in the section VII , in using a

new theorem (lemma III) concerning the connection between the H -determinants and the M -determinants

.
4a

Ve give in this section another application of this theorem in

estimating the variation in the inverse matrix of a triangular matrix

satisfying the conditions (2) with m = 0. We obtain an unexpected

simple and elegant formula (VI ,33).

In the section VII we apply our results explicitly to

the problems concerning the linear system (1)

It may be finally remarked that our results remain with obvious changes valid if In the matrix
5

of (l) the rows and the columns are interchanged , although we did

2v

not care to mention it explicitly at every step

I . The Value of the eterminant Cl

Let Kn he defined by

(1,1) Kn

V. v. V)-t

Vi.

0 , S

' *1-?. »

K =«< K,r

«t

^2.

K

.

0

0

0

o

, o o

^ K

\ \

( n i 3) )

where in the jn- throw all elements to the left of the main diagonal and

on this diagonal are equal to Kn _^ + ±, the next element to the right

is ^n-icy+i n a,nd all other elements are 0. , , . _

r fO

pendent variables . In subtracting the second column from the first we

obtain H\ ^ ' (K

n -^)K n ^ and therefore the following formula, valid

al^o for n = 1. 2:

K S here inde-

>0 %

Kn

= -

( 1 , 2 )

i
5a

Consider

(1,3)

now for n ^ 3 the determinant

4

0 0 O , . . o <4

c, §

n-i o o 9 » , 0

c la 1 0 .

tv-Z* . . 0

o « <5> €> c 9 >9 © 6 4 4 4 4 . JP

» • Pt.

i i i i . « » i o

( n s 3 ) .
We have in particular

(in) T "a. 1

o e.

s.

0

2, i

Developing Tn in the elements of the first line and using the value

{ 1 , 2 ) of Kn we obtain for n ^ 4

X-- 5 T, - ft, 7$ ( VM ,

tt-1 7X

T, T1 n-i % Tt-fO

^ . . * • S

n . % . • S

n_ (

and therefore generally for n il 4

(1,5)

T\v\

0 • 9 S

T\o- £) + —' f\(i4)+-^r\o-^)|

s

n

>».«.

B - 3

Since by (1,4)

= - A (i- is

)

n i -_ <3i 41 s s,

we obtain
where

1

is identically 1 , and therefore finally

( 1 , 6 ) v-2 iti'(s*"°-

fl-3l

If we now put

(1,7)

$, 0 0 3b c \ 0

® #

«> e

sS» ® 9

0 38k

W

V\« 1

1 -

Cl C • 9 ®

1, 1 ® ® & 1 o <s, 110

r . A

this goes over into Tn if the indices of (5 > % , are replaced by

their complements with respect to n + 1. We obtain then from (1,6)

n ft-1

/*«*» *\>s&

or in replacing the summation index u by n + 1 -

Vt&l, .* Vaift^K it ( C-I 1-0;

and finally, if in both products V is replaced by n 4- 1--?L ,
8

Consider now for n i 5 the determinant

°<i -*1 -Mj. . . • -M

'i

-Mv . . • 'Mi

(1,9) <v-

- -m3 • . . ‘M,

• • • • • • •

~ mn - - • o(

• • ”n

If we subtract here the last column from each of the proceeding ones

we obtain *

0 0

m*-TM* V** o

. Q

" N

'l.

.. 0 "M*.

• 0 ~ j

• .

_(«b4 W,)-(«U*wO -K*"0» «****\^ «<„

Here the subdeterminant corresponding to the last element of the last row

is obviously 7 \

(M^ + ) , so that is the sum of T\ (m v +^)

and
8a

+ <x. 0 O 0 - M>

M,-vn M+«» 0

M3 -Wj

(°<,,+10

V

0 - H8s

M„. r

yv, m*. ^iog !%*8 M •* °(

IWj t\»| - (V
This last determinant becomes Tn

* if we put

} (9V =M V

and has therefore by (1,8) the value

K*l Xs I a.aH+1

¥e obtain therefore for n the expression

(1,10)

y\ k= \ X- Ktl

The determinant _0 can be also written in the form

<Y

n -n% -nn n

,

• • - -^n

-*v, °t-> -"V. • •

(1,11) =

a, % ft

• _wW

•

- m„.

» • * •

— ^ "L. • *» . - wv
and we obtain therefore from (l„10)

II . The Matrix A and Its Inverse

We consider now the matrix

Its determinant is obtained from X\ in putting in ( 1 , 12 )

- m;

We obtain

I | n— * /h, = ( irv\ +1) —

j u /

\

r'“ k K- l ( TM+0 ( M + i)

K = £.

and this becomes,, if M f m, = (m+l) n ^-m(M+l)

(M+l) n 1 -(m+l) n 1

M-M

1

M -m (m+l) n ^ (M-m+m(M-!-l)

( 11 , 2 ) 1

M~

ri M ( w + £) - m (rr\\
11

while for m = M we obtain from (11,2) in letting M —> m:

(H,5)

ft * 1

\ ^J = ( l

In particular we see that under hypothesis tti < M necessary and

sufficient for lA^|>0 is

(11,4) m / M

(m+l) n v (M+l)n

Tie assume now in particular

(H,5) m < M

If we introduce the abbreviations

(II, S) M,

_ (n-nf-nM-1

^

in,, ;

we can write (II, 2) in the form

(M-m)|An |= M ( l+nm+mmn

) -m ( l+nM+MM

n ) = M-m - mM(Mn

-mn )

and therefore, it we put
(11,7)

' i

(M-m) I

= m M(Mn - m

fl

)

(11,8)

§ _ M„-

rn M

” M “ W

It follows from (11,5) for n )> 2

V**l ( ^

\ v / |v\ - nr\

> Z(

M - m V- z

-v- z

c

?
12

so that from (II ,8) we have

(H,9)

\ YY\

> -i-

^0

>H^h

' H

It follows in particular if (II, 4) holds §>0, 0^ j In solving (11,9) with respect to m we obtain

( 11 , 10 ) s

< m< —— W) * c

X k4

1 VI 0 ~~ *n m

Q 4 ©41

.

M

(11,10°) i <

i M

s

- £<$ < & *

In order to discuss the meaning of the condition (11,4), consider the

curve

(11,11) y = f(x) X

(l+x) n ;
13

we nave

f»(x) = = n

(l+x) n+1 ' (l+x) n+2

1

For x

^

0 the curve (11,11) has one maximum at x = —_

j- an(l an inflection

o point ^ at x = —=- . n -1

In the diagram 1 the curve (II, 11) is drawn (computed for n = 5) • H"10

portion of this curve from the point 0 to the highest point T will he

denoted as the ascending branch and the

portion between T a,nd x = Was the descending branch .

60

vi1

n - i

f(X) =

1

\
If we assume that in (11,4) m is less than M ; we see that either

14

(II.12) 0 <m <1 , M > _A_ n-1 n-1

or

(11,12°) m M -A- . n-1

In order to find for a given M > -A.. the range of the values of m

' n-1

satisfying (II, 4) we find on the curve the second point Pjy[ with the same

ordinate as the point x = M, then if to Pjyj; corresponds the abcissa

m(M) we have 0 ( m ( m(M)

.

Tie prove finally that from (II, 4) and m ( M always follows

(11,15) m M

n_1 < 1 .

In^deed in proving ( 11,15) we can obviously assume that M is > 1 and

therefore on the descending branch in the diagram but then we have

n-v M

w-1 >

lb

1+ M

and therefore in raising it to the nth power

we see that —-

T must IT

M

n -i

follows ~-j> m, that is

Tie are going now to de

s in the fig . 1 between M and m

(11,13)

termine the inverse matrix

(M) and it

(11, V)
15

of A > if its determinant (II, 2) resp. (II, 5) Is ^ 0. It is sufficient

for this purpose to solve the corresponding linear system

(je, 15) M J= ^ = % (>- •, - :, »)

for indeterminants z., . Put

vhn M

M-tl

(HA6) (v\ it m _ KM l^"

i + M —— ~ ^ ^ ^

v Hr X ^ ~ Vi

h-m * > Then the system (II ,15) is equivalent with

;

( L + H) >. = r

x*r o-( *

4 + • • v V-i) +y

or in putting

S. ~ ^i_'+ » O O .“1

” 0 0 ) ^y

s +

VV'r r* i 1

5»f- v. r”

v f ^-v-

n)
16

(H. 17 ) s. = £L yv

» \ v

(ii,i8) *

r -ZI C

y^_

••y*).

Prom (11,17) f°r = n we have by (11,16)

^

VI * yv-V

(T> (11,19) s~- + 22 * V 1 V-L %

where for m = M, q = 1 the coefficient of JCs is to be replaced by n

Since for m ^ M

1 - ^ f-l _ m- M<jT _ t^-1 vn-M M - no

ire have from (11,19) and (II, 2)

(11,20) 5=. Q±*v %

n-V

( voc ;

*1 (11,21) _ M-m _ (M* I) (mifH)

while for m = M we obtain again (II, 20) in defining Q by

( 11 , 21 °) Q =.

( on -1-0 ( m = H)
17

If we now replace in (II, 18)

^

+ and US6 f0P S the ° X

pressions from (II, 20)

But here the expression within the Brackets is qH 1 and we obtain

therefore

x^= X.°<* U-i)+ (u.--h--v n)j

r vTM

and finally in introducing e<v fpom (Hjl6)

( 11 , 22 ) M* 8 V-

n-V

% ( ^

“

^ j n/

*

Introduce the two following triangular matrices

)

( 11 , 24 )

( 11 , 25 )
18

(11,26)

We see then that in (11,22) the matrix corresponding to the first right

hand side is

q-1 TT _ m -M

TT M+l Un " (mTIF n

while to the second right hand sum in (II, 22) corresponds the matrix

q

11 '1

(q un + En + vn )

.

We obtain

Ch+i)

1

/

E>

> (H+i)

v V

The coefficient of U^is here equal to s~2 "both in the case (II, 21) (M+l)

and (11,21°) and we have finally

A

-i „ mQ n /

(II 27) - —U. + ( H* s (M-hV* /

* *

We now denote by s^ the sum of the elements in the ^ th row of the

matrix (II, 27) and by t^ the corresponding sum in the^ th column. If

we assume first = Mj for the matrices Un and Vn the ^

th row terms

are respective ly equal to

. i -

n

while the “$-th column sums are

%-W ) V 1
We obtain now from (II ,27) ty (Il,l6)

V\oa\ ®» |

( NA }• \f % s ^

““* ”t* H*5 i* M •»$» M 0^

“ r

and this is hy (II, 21) = (M+l) Q . We have finally

(II ’ 28) V

= wir^'

,=

r>»« ( M l

-) q p-~l

\A„1

*

and this remains true also for m = M, as is immediately seen.

Similarly, we have

( M + \)N; V = W

<fc -1-

S-I

-V- 1 -f

_ » 1 <41 +

•l n-v

%

_ K I

and we have

(11,29)

wi

a relation which is also immediately verified for the case m =
19a

A

(III

III. Bounds for the Determinants , depending on Xl^and An

determinant

• — Vr» in

1) M -

- vri

• • -

2. 1 * in

•

e *

~ * “ °< e ' r)
20

will be called an M -determinant if all diagonal elements ^are positive

,

all elements off the main diagonal, -mr\ ^

%v) are not positive and

the determinant M as well as of all principal (coaxial) minors of M of

all degrees are positive

.

In what follows we will have to use a theorem given by the author

in 1957 and which will be formulated as Lemma I. If we have for

the M -determinant (111,1) and a determinant H = ) h ^ n of the order n the inequalities

(in, 2) t UN, ,

1 Vv 'bo.

then H ^ 0 and we have

(III, 3) Hi > M

f ( H)

,

(M) respectively denote the matrices of the determinants

II and Mythe inverse matrix of (H) is majorated by the inverse matrix of (m)

(III, 7°) (H) -1

<$C. (M) "-*

Suppose now that in the determinant _l\ given by (1,9) all <X are

b v posi ive and non negative: Then it follows ”rom (1,10) that

n is a monotonically decreasing function of all m M and from (1,12)

that M is also monotonically decreasing in all . Suppose now that

for a certain set of values of m , M v the determinant ^ 0,
20a

Then replace the small nr. and I/f corresponding to certain rows

V

h) \ by zeros Ithe determinant _0_ n cannot decrease and remains

v 1

>• • • •; J therefore positive; hut then _fLn becomes equal to the product of

o( v

c^

v with the principal minor complementary to the set of

1 V * * \

indices \\ -o , Therefore all principal minors of _0_ ^ are

positive. We see that l~h n is an M -determinant , if the conditions

(III, 4) are satisfied and f\ ^ 0 . But now we can easily deduce

the following theorem.
21

A. Consider the set_____ of all — determinants A = S 3. I satisfying « 1 - - the conditions

' \) 4 1= tV

(m,5)

hf >

where are n -a- given — ^positive constants and m , M 2n-2 given non-

(\

3 fi ~—

negative constants . Then necessary and sufficient in order that all

determinants A of this set d o not vanish is the ineq uality Ji > ° If this condition is satisfied, we have

(HI, 6) I Aiin.

Proof: Since in the case _fl>0, _PL is an M -determinant , the suffih n ciency of our condition follows immediately from the inequality (III, 3)

mentioned above

0 we can take obviously a^

and V = V op

a^

v

= according as ^ or v and obtain a vanishing determinant A

satisfying the condition (ill, 5 ).

Suppose now Si^O, then it follows from the form (l,9) of jfL-^ that

_fLn becomes positive if all m are replaced by zeros. There exists

therefore such a positive t < 1 that Jf^vanishes if all m^ in (1,9)

are-replaced, by t m^ , but then we obtain a vanishing determinant A of

our set in taking a, =©<. and a _ = t m. or a

(A.jLV > ^.V p- pLV

M, according

as or V < . The theorem A is proved.

In specializing the matrix of j[\. to and in assuming that in particular 0 <; m <; M we obtain . immediately from (II, 2) and the theorem A:

B . Consider for two positive constants m, M, m CM the set of all

determinants A = Jof the n

td order for which

(Iiuplajll }

m
22

L hen nece ssary and sufficient in order that all determinant s A of_ thi s

set do not vanish is for m <_M, the inequality

(III, 8)

no ^ H

( vn 4 N\)

and if this inequality is satisfied we have for each determinant A of

the set

(HI, 9) I A \ > \hj-- ( H r m

m - TM Lo + m)"

If m = M, necessary and sufficient for all determinants A being / 0 is

(III. 8°) m < JL (M = m)

My ~ Te have, if (ill ,8°) is satisfied ,

(III, 9°) I A ! = \

1

( t“-(n-s)nr>)(!+m)n

’ ( no = M),

ffrom the theorem B

C . Let A - (a^

we can deduce the following theorem.

__ } ( juu, y = l,.,,,n) he a matrix satisfying the

conditions

( 111 , 10 )

where in, M are two constants with 0 < m < M

Put

( 111 , 11 ' ,v VI rrt ln -

S ( w\ m) - - Jj; )
23

then all fundamental roots of the matrix A are contained in the set

of the—n closed circles described around the elements a with •—— fir —

the radius &(m,M) . The value (ill ,11) of S(m f M) is indeed assumed

if A =A 5

) . »\' Proof:

Let X be a fundamental root of A so that the matrix ^.E-A is singular.

Put Min | = o< , we have to prove that °( 4 Sfw.hl) • If o< = 0 there

is nothing to prove. Suppose > 0 and consider the matrix

- (V);

. r this matrix we have

1 b

f. r

' - L

1 Vv ' = ^ >L 1 ^

)

therefore we have by the theorem B

and therefore

(

m. M

+ ot 4

1 + *< wi l/n * h’ m '

•/*» M -) $

M VY1

*V>

4 M — m

W> - i (

vvy, M)

IV. The Bounds of the Matrix rX

T\

For an n dimensional vector (x^

. .,xn ) the Ho Ider norms

corresponding to the exponent p > 1 Is given by

(IV, l) I 1 Ip* fy U/+---+ IxJ*

1 (?= ah
24

We will only use the three cases corresponding to p = 1,2, oo*.

We have "between these three norms the following inequalities

(IV, 5a) 1? Soo

^

(IV, 5b) 1 ? \oo I ^

(IV M) III, 4 ,

which are Immediately verified. The left Land inequality (IV, 5c) Implies

the well known inequality between the arithmetical mean and the arith —

metical mean of the squares . If A = (a ) is an n x n matrix we define

its norm corresponding to the exponent p(l|p4®°) by

(IV, 4) \ A

and denote it by | A \ p

.

We will use here too only the cases p = l,2,o0.

In applying to the definition (IV 4) the formulae ( IV , Is. ) , (IV,yo)

and (IV, pc) we obtain immediately

(IV, 5a)

eO (rv, 5b)
o C

-J

For p = 1 ; °° the expressions of \a ( ^ l^L© are easy to write down*

we have as is well known and very easy to prove

(IV, 6a) ph-MaxXla \

V r* r

(IV, 6b) UU = Max Ella. 1 . r* r

A-g to (a\ 2 , its expression is irrational, \A\ 2 is the square root of

the maximum fundamental root of the symmetric and non negative matrix

A A*. Since the direct computation of |A

| ^ is i- n e105 ^ cases difficult

we prove in what follows the following estimate for |A)g:

i

I Lemma II . ¥e have for any matrix A

(IV, 7) ~ Max ( l a]j

, | A| ^ )£lA| 2 ty \A|! |Ai w ,

The first part of (IV, 7) follows from (IV, 5a) and (IV, 5"t>) • To prove

the second part we introduce the notations

$ *2-

r v

^ \ J ~ ^

I v I ( h V ” l - • V1) •

The sum of the moduli of all elements in the M th row of AA* can he

If estimated as follows
25a

(IV, 7) follows new from the theorem of Frobenius that the modulus of

each fundamental root of a square matrix does not exceed the greatest

sum of the moduli of the elements of this matrix in different rows.

/srl if we apply now these results to the matrix discussed in the

r% section II, we obtain from (11,18, (11,19), IV, 6a) and (IV, 6b)

(IV, 8) ! I
26

and from (IV, 7) and (IV, 5a)

(iv, 9)

('m+iV

h

\ O

'

In combining these inequalities with the result given in the section III, we obtain the following theorem.

D . Let A = (a^,0 ) be a square matrix of order n satisfying to the

conditions (III, 7) and let (III, 8) be satisfied, then we have

|A-1| (p = l,2,oo)

l^nl

and therefore for any vector %

dv,io) (?*^«).

Proof . It follows from (ill, 5°) of the lemma I at once in virtue of

(IV 6a) and (IV, 6b) that

(IV, 13) ?

a

VHrt

further, the matrix A~^(A~-^)* is majorated by A^(A^) . Since

therefore by a well known theorem of Frobenius, the maximum modulus of

a fundamental root of A~-*-(A"l)* is majorated by that of A^ , we

have

(IV, 14) |a"|, ^

\ 4;^- j

and therefore by definition (IV, 4) in putting A^ =

and this is equivalent with (IV, 11)
A- I V. The hounds of the matrix £1^ — Zh

We denote by A

<0 the matrix obtained from (II. l) in replacing there

m everywhere by zero

.

1 - N\ . . ,-M

(v,i)

Tie will prove the following theorem:

I f we have o^mOl and (ill

,

8 ) is s atisfied, then the matrix

A"V\ - &

W- 1 .is a non negative matrix and we ha.ve

|£'-AeVI

| * ( H-Nf'

1

' Trl^ei =

Itvftn I M

(V,2) l<-

Jo)-

Proof

:

Since tislies the inequalities (III ,d) it ioiiows irom

(ill, 3°) that is non negative. Denote by and tj°) the

sums of the elements in the ^

th and in the th column of and by — — a^S

s^ and t^ the corresponding expressions for the matrix **

¥e have

(V,3) ( 0 ) lu. ^

rr \

-r jf

It follows then from the formulae (11,22) and (II, 23) that s^ run

for ^

= n, n-lj..,l through the same set of values as for V = 1 , The formula (II, 29 ) can be written in the form

v*-y o>-l

(mdl

(v ’ 4) t v= IAJ

" IaJ
28

Since becomes 1 for m = 0, we have (V,2) and E is proved.

Discussion of I Ip. We prove first that t^

goes increasing with V as long as the condition

(V,5)

is satisfied. Indeed we put

(v,6) k «

tv*' v = (M — t-C u.i

In solving the 3 inequalities

(V,7) k s 1

V

^

with respect to | , we obtain correspondingly

,v - 8) Ki^ = (i.{i)(i

The inequalities =. l(~$ = ,1, . , , ,n) are obviously satisfied

in virtue of(V,5) as long as 'V — n-1. We have therefore in

this case, in using (ll,7)

c vO=

(

,
(28°)

(v ’ 9) u;-Ari=(i+ nf^=(ltMri-i_ ,

f t- i <P«l,«jK,l*i-E) >

The condition (V,5) can be written using the notation (II, 7)

c ^ JXL

® ~ M

and this is in virtue of (II, 10) certainly satisfied, if we have

M Mn t 1,

(v,10) (M + l) n - n M - 2 i 0.

P;

(:

si

s

We

0

[t

V
29

The condition (V,10) becomes for n = 3: M gi . 5321, For n 4(V,10)

is in any case satisfied if we have

(V,ll) M £ i-i5 (n H).

Indeed if the relation (V,10) is satisfied for a positive M it is

satisfied for any greater value since the coefficients of all positive

powers of M in the left side expression are greater or equal 0. To

prove the sufficiency of(V,ll) it is sufficient to prove that

(l + ^2.)

n

^3*5 (n ^ 4). But here, the left side expression is

monotonically growing with n and this inequality follows therefore from

S B 4

(^-i = 3.75. • • >3-5

Suppose now that we have

(v,i2 ) i--jy k\\ 1=1 Si ^

.

Then we have, since 0<m<-JL-,

n -I

(t+mf'-l < C+_^rLzi_ t I + “') = (e-l)(h-l), —m = _Jl_ J

(>4 mf = | + 1.72 Sum ,

( i + mf~'t- 1 hA

( -a 1-7Z 9 n m + i - !

and therefore by (V,'l2) ( K ^ ^ I ^ l * 1 Z, 0 y> 4- 'pp •

For n ^ 4 we have in the case (¥,12) since the inequality (V,ll) is not

verified,

1 <_JL

M ^1.5
30

and. therefore finally

(V,13) (1 + m) n ~ V- ) 6 2.39 n m

On the other hand, since M <3 -k-.i , (l-tllf S (1 -1- 4. e"

1 ‘

’ = 4.481689

n n

and therefore from (V , 2 ) and (V,3)

(V, 14) I i \C,7Z» ^ (pl / 0o. I I- - j H>3)

To obtain a lover bound, ve take in (V,4) y= 1* W© obtain, since

0 <, I j 4 1, t 1 -ti^°b l.UglV.lI .> .(.h-lln v

n

_jn_

!A, IAJ IM

and therefore

(von | a,

< p

,

„

t+).

To obtain the exact value of | jp if(V,5) is not satisfied,

ve return to the inequalities (V,8) equivalent to (V,7).

Denote by n 0 the smallest integer betveen 1 and n such that ve have

(V,16)

1 - I ^ I * (ih^flj,

The parts of this inequality implying GT* or rp must be disregarded,, that

C '“A

is to say, this inequality reduces to j > ^ or n,

t0 *

f (c), _ < tn, ^ im or n. Max ( - ty') =

°n 0 k

( 0 )

O n

, — -o

Then ve see at once that

1 and

and therefore

(v,i7 ) Ia

a

-Aa |a„| 1 ' J)

For m = M ve have nQ = 1 and therefore
51

(V,17°) ! a;'-4(c)~\

l-Ch-O

( msH

VI . The hounds of A‘

-1 _ A

(0)-l

In order to obtain the theorem corresponding to E for a determinant

A satisfying the conditions (III,7 )j (III, 8) of the theorem B, we prove

first the following important lemma which generalizes considerably the

relation (111^3°) of the lemma I.

O

Lemma III. Consider n positive numbers to and 2 n - 2n non

negative numbers

m^-v ) Vv p-P - i; -~j r;)

such that the matrices

(VI,1) M =

(VI, 2) M

s -"V 6 * - - * “

“*\Ai * * •"*

fe a ft ft

* * * CK

/ -<i

• * * ~ >^10

0^ 2, *

i *

, ®

* * « \- w«. — ft « ® iu

. /

j

are M-ma trices . Consider n constants A ( *. = l, 0 „ OJ n) such that

(VI ,5)

v- 'h"

I \t = —j

2 and 2n" - 2n constants b^

v ( ... n) such

that
32

(VI, 4) <V-»I “ %V >

I * t*-*

V

5

/

tv

v=

and forrr the two matrices

(VI, 5) A = f ^

i

\.\ ^X • » • ^vn

• • » • V b

n X - • • !

ft I , . . a,vi \

(VI, 6) A

(0)

Ai « • #

a a. h i. * * • ^n/

)

then we have

(VI, 7) A

-1 A ( 0 ) -1

<<* M

1 M

Proof : Ve can write M = P - T, M

ho)

( 0 ) _

(0)-l

.( 0 ) P - T

vw/ where the matrices T,

have in the main diagonal zeros and off the main diagonal respectively the non negative lements m +4 in , p-v {*''*

the diagonal matrix / o(

while P is

0 \

(VI, 8) P = 0

\ \

v
32a

We can develop the inverse

in the following way

of P-T = P(E-P X

T) (E is the unit matrix)

qO

(VI, 9) (P-T)

(p-lip)k p-1

The convergence of this development anci the validity oi {\f±. , ^

)

foil s
33

easily from the fact that the determinant of the matrix P-tT does not

vanish for J tj % 1 as follows immediately from the lemma I. The corresponding development holds also for (P-T^ 0

^)

- ^ and we obtain therefore

(VI, 10) M

-1

oo

_m( °) -1 = J^P"1

^ - (p-1t(°) ) k

J

P" 1

K-*\

The elements of P ^T are here ***.,. or zeros and those of P ^T°

are or zeros, therefore all elements of the matrices

(VI, 11) (

P

-1T

)

k - (p-1t(°)) 1<: (ic — 1,2,...)

are polynomials in hr and twith nonnegative coefficients. Depb* note by Q, the diagonal matrix

Q. =

A,1 ^ \ ^ 0

' 0

and write /

A=Q -S , A v ' = Q 1 - S

( 0 ) ( 0 )

where the elements of S are b., . or zeros and those of a. HV f zeros . Since therefore we have

L'U

Q~^S « P _1T , Q-isCo)^ P“ 1t(°) ,

we have

A' 1 = (Q _1s) k Q

_1

, a(°)

(Q- 1s(°)) k

qr

1

K-0 K-o

(VI, 12) A" 1 -a(°)-1= 21 \

(Q-1

S) k - (Q^1 s(°)) k '\

«*.0 'v J

Q

-1

But now the elements of

(Q

1S)

k

- (Q“ 1s(°)) k

are obtained from those of (VI,ll) in substituting there

"

" V

instead of
34

and

and (VI, 4)

b/4, -y instead of Here we have by (VI,3)

A

l 4 | Vv <Vv \ 4

I" "V-

;

1 —v“ r <Xf.

and therefore since the coefficients in (VI, 11) are, as already mentioned, not negative

(Q

- 1S) k - (Q

- 1 s(°)) k << (?

_ 1T)

k

- (P 1

i

, ( 0

' ) k

and

(VI, 13) 2-J(Q

1 S) k -(Q-1s(°) ) k

J

<<C^ ^ ^P"' 1 T) k -(P“1T(°) ) k

Ksl J

K*l

But from (VI,13) in virtue of (VI,3) it follows that the development

(VI, 12) is majorated by (VI, 10) and our lemma Is proved.

Under the conditions of the theorem B in v the section III, if the

inequalities (ill, 7) and (ill, 8) are satisfied, we can apply the

lemma III in replacing there M by by . We obtain

P « Under the conditions of the theorem B, if the inequalities (ill, 7)

and (III, 8) are satisfied, we have

-I -HeVl

(VI, 14) \K'~h‘"\

?

<\ a)- a7"

p= m* * ( ,* (?

^

where the values and the estimates for the right side expression in (VI, 14)

are obtained from the formulae (V,9) to (V,17°)

The lemma III can be applied in many problens similar to that solved

by the theorem P. Por instance, in the theory of the solutions of linear

equations the following problem has to be dealt with, although its complete

discussion is usually avoided:

Consider a "triangular 1 system of linear equations

(VI, 15) a.._A.= ^ t. ----a) v t*- )
55

where the coefficients a are only approximate values to the "true

values b

c

(VI, 16)

“(A. V

Suppose that we have generally

how far the solution of the system

( VI >17) Z>

Vr

jia^

'

Is influenced if the b ^

are replaced by a^ ? If we denote the matrix of (VI ,15) by a(°) and that of (VI ,17) by

A, the question can be in particula? answered by giving estimates of

[A' 1 - A(0)-1 \ p

(p = l,oo) .

Suppose that we have generally

(VI, 18)

and consider the matrix

(VI, 19)

VfA ~ 1

;

1 Vv ' ~ *V (y. >n- ; ^us

At K„.,) =

I -M, —~M|

0 I ... -Mi

* % *

\ o o . . , i.

where all elements to the left of the main diagonal are zeros, while

all elements to the right of the main diagonal in the ^

th row are equal

to M

(M

' We obtain from the lemma III at once the majoration

r\

To obtain the inverse of the matrix & (M^, . . . ,Mn _y)

consider the

system of linear equations

(vi, 20) x w - ^

= w. C /«'-’/ --v

n).
56

To solve it explicitly ve put

(VI, 21) s„ =

K*fi (VI, 22)

Then (¥1,20) he comes

(

• • • j n )

C jtA. - \j » , . -J - D

(VI, 25) x

r

t

-X

II S

|((+l

+ U

(4 ( f4 =

(VI, 24) s

II -V 3 S|qTl + U

(M

((V = U

where Sn+1 = 0

Dividing (VI, 24) by P we obtain

. . ,n)

5 . ,n)

R

t Jid

and therefore

>

introducing this in (VI, 25) we have

(VI, 25)

Ve obtain therefore for-tlie inverse of our (M ,

. . . ,Mn _]_) , in c

V -rr TP oy the unit matrix,

(VI, 26)

rl

A(Mj_, . . .,Mn _x) = E + D T,

where D is the diagonal matrix

(VI, 27)

D =

and T the triangular matrix

0

Mm Pan o ting
37

The expressions

(vi, 29) A(W ••j'VT*) - A(Hw

». v

Hn^')

obtained from (VI,26) to (VI,28) is of course rather unwieldy; however

we shall obtain for its norm corresponding to p = 1 (cf IV, 6s.) a very

simple and elegant expression.

Indeed, if we take the sum of the elements in the "V th column of

A (M1

,

. . . ,Mn _-L)

“ 1 and denote it by t^ ,we have

•t' -i+-pZl H P

= 1+ A M TV (i + %) . v r

(ltl /««i ^ x>*i

If we write now tb + -|_ out, we obtain

and in comparing this with the expression of t

1 we see that we have

tr - =

' 1^, r M ) t

1 and therefore t!l"'T V

7 V

(VI, 30)

-v-l

t

1 =

V 7\ (1 + Mk ) KWe obtain now for the sum of the elements in the v th column of (VI,29)

ti = 0, t 2 = £

K* \ i K= l '

In multiplying this by 1. + t we obtain

7V ( >, M„t«) -TV (w*0 - 4 S"f MH0 ; K- l and comparing this with t^+ ]_
V-4

38

(VI, 31) -t = ( I+M -f Ot, + * AC l + Mj . > * K =\

Therefore, t ^ is monotonically increasing with ^ and we see that the

norm of (VI, 29) corresponding to p = 1 has the value

n-i n -

and obtain therefore

(vi, 33) I A~

I-A

CoV!

| 4 'TV (l + H v

+fe)“7T (

1 v - 1 v= t

as the solution of our problem. In applications it may be better to use

the recurrent formula (VI,3l) . If all M.. have the same value M, the

expression (VI,26) coincides with that obtained in IV for under the

hypothesis m = 0. But in this case we see from (11,28) and (11,29) that

the row sum run through the same values as the sums of the columns. We

obtain therefore in this case the expression

(M+l+t) n " 1 -(M-l-l) n “ 1 = (n-1) e (M+1+0 t ) , 0<9<1,

as the norm of (VI, 29 ); both for p = 1 and p = oO •
38a

VII . Linear s ystems with a Nearly Triangular Matrix

The results of the preceeding sections give the means to discuss

the following problem concerning the system (l) in the introduction

under the conditions (2) and the "triangular" system.

(VII, 1)

= 1,—,«) J

with the matrix A K •

. In discussing this problem we can obviously

ass ume tha t

( ~ 1 )

* • • * 1
Then the difference "between the solutions (l) and of (VII,l) is given

by the vector

;

and the norm of this vector corresponding to one of the indices p

does not exceed

1,

|a' 1 - A

(0) -II

lo\ p

(p = 1,00),

and can indeed for suitable choice of the vector ^

attain this limit.

Therefore the norms I 1 measure the error committed in replaci • 1 P

the system (7) by (VII, l).

For an f > 0. M > 0 being given , how small must m ) 0 be taken in

order that we have

(vna) (a

" 1

- a(°)-1

| o i * (p = i, oo)

•

If we introduce the quantities \aa and § corresponding by (II, 2) , (II, p) and (II, 7) bo m and M, we obtain from (V,9) and (VI,7) the condition

(1+M) n_1 1 (= £

(VII, 3)

(i (\AJ4 1-*)

as long as the condition (V,5) is satisfied and therefore certainly

as long as M > —2. ( n ij.)

# On the other hand we have by (2,10)

(VII, 4) m

© A

M

and from (VII, 5) and (VII, 4)

(VII, 5) rr\ 4
40: It -..ill te therefore sufficient for (VI 1, 2} to take

(VII, 6)

1 R solve our problem, for instance, if we have M ^ ( n ^ 4) or

U > .5321 (n = 3) . For small values of £ obviously, only the first term

In the denominator is essential and we have

(VII, T) m »K(n,M)€ , K(n,tl) =

1 —

Mn(l + M) n_1

The tables I and III give the values of K(n,M) for a set of integer M

from 1 to lOand some values of M > injz . Ve have obviously

» n

(VII, S) K(n,M) < - -J-. - - M^n

The bound in (VII,6) is obviously the "best” under the condition

(V,5)save that the factor I of ^ in the denominator could be replaced

M

by an (unknown) fraction of it.

If il< 1 »5 , the use of the general formula (V,17) is very cumbersome;

n

ve can however obtain a good working limit for m in the following way. If

the inequalities (2) are valid for an , then are also

n

satisfied if M is replaced by , but then the limit for m obtained

for M^ n ) is also sufficient for our M. Ve obtain therefore in this case

for m the sufficient condition for (VII,2).
40a

(VII, 9) rv> 4 no ( -

The table Ilgives the values of K(n,M( n

)) fop n = 1 , 2 , ...

,

50 . The expression fop m 1 can he written in introducing the value... of

Mn

(n)

a:

3_

2n

6

(1 + i.O_)n-l ((l + l^)n_2-5) + ((1 + L5)n
We observe now that for any oositive ©< and positive x the

expression (l + —

)

lj meno : / -mil ' increase and tends to

e

* if the positive n increases monotonically to oo . Indeed , if

we put u = — , take the logarithm of this expression, differentiate

. , . , , , , , . x ( u - <x , it with respect to u and multiply by n. , we obtain — i~+~u — log ( l+ux)

but this expression vanishes for u = 0 and decreases for positive

u since its derivative is

-xu

(l -f* u x)

(2a + x + a x u)

We see therefore that (l + ~5-) n ^ t e~*5, (l + ie5.) n ^ e~ * -

and our bound for m can therefore be replaced by

^ . e 1.5 (e l.5 „ 2 .5) + (e l.5 _ !.5)t

05.93 + 1.99 )n

we can replace therefore the condition

condition

to in ' ,, by the simpler

(VII, 10) rr.

i 211 = — 1 __ * * o n 7TV 2€

On the other hand we obtain at once the solution of our problem

in the case m = M <. _JL_ from (V, 17 )

.

n-1

(VII, 11) (m = K^_r

r) .

This is an "exact” condition, while the condition (VII,10) is

nly an estimate. We see however, in comparing (VII, 10) with

(VII, ll) that the bound in (VII, 10) cannot be improved by a factor^ 6.
42

From the condition (VII, 10) we can finally derive the inequality

( 7 ) of the introduction. Indeed, if i- positive mo^ 2^

- s S 1 A/erh* Te

solve (VII, 10) with respect to (z ,

<z =

6 n m _ 1-2 n m0

and obtain therefore in applying (VII, 10) for m = m0 the inequality

|

A-1 - a(°)- 1| h n

l-2n mo

vhich holds for a

¥e replace here m0 hy m.

11 positive mc ^ and gives the inequality (?)
Bib liography

til Ostrovski * A. Uber die Determinanten mit uberviegender Hauptdiagonale

Comm. Math. Helv. 10* (1937) P. 69 - 96 .

[ Ostrovski* A. Sur le conditions generales pour la regular! te des

matrices* Rendic . di Mat. e.d.s. applicaz. (V)* 10(1951)*

p. 156-168.

£5] Ostrovski* A.. Sur le matrices pen differentes d'une matrice triangulaire * G ,R

255 ( 1951 )* p. 1558-1560.

L.4J Taussky-Todd* 0. A recurring theorem on determinants* Amer . Math.

Month. 56(1949), p. 672 - 676 .

|ja] Stein* P. and Rosenberg* R. L.* On the solution of linear simultaneous equations by iteration. The Journal of the London Mathematical Society*' Vol. 25 (1948), p. 111-117.
44

Foot Notes

1) This condition (3°) is already contained in some results of my paper

in , and can he also deduced from a well known theorem of the theory

of determinants discussed in tn .

2) Some of the results contained in the section I - IV have heen published

without proof in my note £3^ .

3) For M = m, 8(m,M) becomes (n-l)m, but in this case our result is

obtained in Gerschgorin ' s theorem cf. I4Q . From the theorem c the result given by Stein and Rosenberg in jjjsQ as

theorem III follows immediately.

2a) This formula can be also obtained from the formulae given in the

proof of theorem III in £3a) , p. 113.
Table I

K(n,M)

45-

M n=3 n=4 n=5 n=6 n=7 n=8

1 .0625 .0114 .1[2)1240 .(3]1548

2 .0111 .1 2 1103 A 3 1106 .4 1115

3 .(2;1347 .1 3 1193 A 4 1116 . (6 1719

4 .2 1143 .1 4 152 6 A 5 1206 .(7 (821

5 1694 .1 4 1182 A 6 1498 .(7 1138

6 1378 .1 5 173 6 .1 6 1149 .8 1304

7 .{3 1223 .1 5 1336 A 7 1522 • (9 (815

8 .(3 1 140 .1 5 1 168 .

7 1207 .(9 1255

9 .4 1926 .1[6;1903 .1 8 1900 .(10)900

10 .4 1636 .116)1515 .1 8 1424 .(10)351

.(3) 130 .(4)31

.(5)126 .(6jl4o

. (7)448 .(8)280

.(8)328 .(9)l31

.(9)383 .(10)106

.(10)619 .(11 126

.(10)127 .(12)199

.(ll)315 .(13)389

.(12)900 .(14)900

. (12)290 .(1^)239

M n=9 n=10

1 .(5)778 .1 5)193

2 .(7)155 .1 8)172

3 .9)]L75 .

10)109

4 .11]1524 .1 12)210

5 .(3.2 1295 .< 14)821

6 . (3.3 1258 .1 15)526

7 . (3.4 1311 .1 16)486

8 .(3.5 1 48o . 1 17)592

9 .(3.6 1900 .1 18)900

10 . (l8)1198 A[18)164
n

5

4

5

6

7

8

9

10

11

12

15

14

15

16

17

18

19

20

21

22

25

24

25

26

27

28

29

50

51

52

55

54

55

56

57

58

59

40

Table II

M = M

(n) = 1^5

n

(l+M) n -n M-l K(n,M) n (l+M) n

-n M-l K(n,M)

.871 .255 41 1.865 .00466

1.074 .154 42 1.866 .00454

1.215 .0866 45 1.869 .00442

1.515 .0625 44 1.871 .00451

1.595 .0480 45 1.875 .00420

1.454 .0587 46 1.876 .00410

1.504 .0525 47 1.878 .00401

1.546 .0276 48 1.880 .00591

I.580 .0240 49 1.882 .00585

1.610 .0215 50 1.884 .00574

1.655 .0190

I.658 .0172

1.677 .0157

1.695 .0144

1.710 .0155

1.724 .0124

1.756 .0116

1.748 .0109

1.758 .0102

1.768 .00965

1.776 .00914

1.784 .00869

1.792 .00827

1.799 .00789

1.805 .00755

1.811 .00725

1.817 .00694

1.822 .00667

1.827 .00642

1.851 .00619

1.856 .00597

1.840 .00577

1.844 .00558

1.848 .00540

1.851 .00524

1.854 .00508

1.857 .00495

1.860 .00480
H

CM

K\-=f

LP\VT)

t—CO

CT\ O

rH

CM

K\-3*

in

MO

t—

CO

OMO

H

CM

K\-=r

UOMD

C'-CO

G\O

4 7

Table III

K(n,M)

M n=3 n=4 n=5 n=6 n=7 n=8 n=9

.254

.181

.134

.102

n=10

. 0866

.0328 .( .0399 .0196 .0101 .13197

[2)533

.117 .0438 .0180 .1 2]>788 .1 2;>359 .1 2 1169

.0718 .0241 .(2]1891 . 1 2 1349 .1 2 1142 .1 3 >592

.0464 .0141 .(2 1470 .1 2 1165 .1 3 1602 .1 3 1224

.0313 .00864 .(2 1260 .1 3 1825 .1 3 >270 .1 4 1902

.0218 .00548 .(2 1150 .1 3 (431 .1 3 1127 .

4 1382

.0156 .00359 .(3 1894 .1 3 1233 A 4 1623 .1 4 1169

.0114 .00240 .(3)1 5-48 .1L3J1130 .

4 1316 .1[5J1778

n=40

n=15

.01'57

0121 .1 2 >137

(2 1289 .1 3 >167

(3 >809 .1 4 1242

(3 >252 .1 5 >399

(4 (848 .1 6 >729

(4 1305 .!i 6j1146 . 4 1116 .1 7 >317

5 1 462 .1 8 >743

(5)1 193 A 8 H86

n=45

n=20

2)

439

3)

188

4)

112

n=50

4) 604 .1[4)224 .1[5)841

6)112 .1 7)JL80 . { 8)290

9)299 10 1217 .1 11 H57

11 1114 .1 i3 >395 .1 3.4 1137

14 1613 .1 15 1106 .< 3.7 1184

16 1 449 .1 18 1 409 .1 20 >372

18 1436 .1 20 1216 A 22 1107

20 1545 .! 22 >153 A 25 1 428

22 1856 .1 24 1140 .1 27 1228

23)1165 .1[26;1162 .1[29;1158

n=25

138

281

'793

681 .(8)118 .1 lo)

658 .10)597 .1 3.2

)

721 . 11 357 .1 13

886 .(12 248 .< 15)

) 121 .(13)197 .1,l6)

) 182 .(14)178 .<[17)

30 n=35

469 .(3)166

439 .(6)697

570 .(8)413

957 .(10)331

)204 .(12)354

543 .(14)494

177 .(16)879

695 .{17)195

)322 .{19)525

5 173 . (20)169
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

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