NATIONAL BUREAU OF STANDARDS REPORT

2353

TABLES OP INVERSES OP FINITE SEGMENTS

OP THE HILBERT MATRIX

by

I. Richard Savage and Eugene Lukacs

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

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

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103-50-5119 11 March 1953 2353

TABLES OF INVERSES OF FINITE SEGMENTS

OF THE HILBERT MATRIX

by

I • Richard Savage and Eugene Lukacs

PREPRINT*

The preparation of this paper was sponsored by the

Naval Ordnance Test Station, Inyokern,

The publication, reprint

unless permission Isobta

25, D, C. Such permlss!

cally prepared If that a

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

it, Is prohibited

rds, Washington

las been specif)*

for Its own use,

TABLES OF INVERSES FOR FINITE SEGMENTS OF THE

HILBERT MATRIX*

by

Io Richard Savage and Eugene Lukacs

National Bureau of Standards

1 0 Introduction ,, The purpose of this oaper is the construction

of tables of the inverse of the matrix ]| 1/ ( i+ j -1 ) !L • - 0 These tables are useful in estimating the mean value function of

certain stochastic processes „ The same matrix occurs also in

least square theory when an integral is minimized instead of a sum.

These exact inverses may be used in the testing of processes on

the inversion of matrices on high-speed automatic digital computing

machines for these matrices are ’’ill condi t ioned” 0 Tables of the

principal latent root and vector of these matrices may be found

in [4]

.

2. Inversion of the matrix ,, In the following we consider the

matrix

Sn =

i » J 4 ,n

(1)

and propose to find its inverse S"^ by applying the following

theorem due to A„ Cauchy [1]„

Theorems Let aq ,

„ „ „ s,&n s> bq 5 o<,„ s b n

be 2n numbers and consider the

determinant whose elements are of the form ( a i +bk ) =

^- (i,k=l,2, o.,n) . Then 1 „ „n

1 a i+b k

i,k=l

( a j" ak) (b j“bk )

1 . „n

II ( a

i +bk ) j,k J

o ( 2 )

This theorem* as well as an indication of its proof may also be

found in [5] page 98 s problem 3° Clearly the determinant of Sn

as well as all its minors can be evaluated by (2)

„

We denote

by the minor of the element in the i~th row and j-th column

of the determinant A n of Sn „ If we write for the element in

The preparation of this paper was sponsored by the U, S. Naval

Ordnance Test Station^ Inyokern 0

- 2 -

the i-th row and j-th column of then

(-D

1 + j Ai J /A n ( 3 )

Applying (2) we find by an elementary computation

n-1

and

“n

k

^n = k=l

n-1

n

n (n 2 -k2

) n " k

k=l

r+l-l) l(r + j-l)

It is easy to show inductively that

k

2

) n ~ (n+k) t

n

n l

n — ( n

2 °k2

) n " kk = 1

k=l

n-1

k=l

k \

[ ( i~l) .' ( J ”1 X 1- * ' (

;.n -i>..t(n-j } l IJ.lI

I I (n+k)

k=l

(4)'

i+j-1

(5)

( 6 )

We obtain from (3), (4), (5>), and (6)

ij _ ( ^l) 1 ^

(n+i-1) l(n+j-l) t n

[

( i»l) l( j-1) i] 2 (n»i) l(n-j) l (7)

The inversion of the matrix (1) was studied by R. Collar [2],

[3], the inversion formula (7) is given in a different notation in

[3]. The proof given above seesm however, to be somewhat more

elementary than the proof in [3] and applies also to the more general

matrix inverted by Collar,, Prom (7) we see immediately

oi j = (n+i) (n+j)

n+1 (n+l-i) (n+l-j)

S n+1 >

J

n+1

S j > n+l_

n+1 (n+j)

for i , j = 1 ,

. . . ,n

(2n+l) l ( n+ j ) t [n!( j-1) l]2(n+l-j) !

.(3)

for j = 1,2,..., (n+1)

- 3

-

The last formulae can be used to compute the tables of the

systematically. Tables were computed by means of (8) for

n = 2(1)10 and 1 < i < j ^ n. The tables were computed by Mr.

Edwin L. Grab and checked from the proofs by direct multiplication

on SEAC , the National Bureaupof Standards Eastern Automatic

Computer

.

REFERENCES

[ 1 ] A . Cauchy, Exercises d» Analyse et

pp » 151-159 ^ Paris 134-1 = 2e serie XII, p . 177

•

de phys. math, vol . 2,

Also Oeuvres completes

[2] A. R. Collar, On the Reciprocation of Certain Matrices.

Proceedings of the Royal Society of Edinburgh, Vol. 59*

pp. 195 -206 , (1939).

'

[3] A. R. Collar, On the Reciprocal of a Segment of a Generalized

Hilbert Matrix. Proceedings of the Cambridge

Philosophical Society , Vol, 4 11 - 17 , (1951)°

[

L|. 3 R. A. Fairthorne , J„ C. P. Miller, Hilbert's Double Series

Theorem and Principal Latent Roots of the Resulting

Matrix. MTAC , 3* pp . 399-400, 1949=

[5] 0, Polya, Go Szego, Aufgaben und Ldhrsatze aus der .analysis,

Vol. 2. (Grundlehren der math. Kiss. Vol. 20) J„

Springer Berlin, 1925= Reprinted by Dover Publications

New York, 194-5 =

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THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

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of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950.

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J1J1