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APPROXIMATING THE INVERSE OF A SYMMETRIC POSITIVE DEFINITE MATRIX

Gordon Simons and Yi-Ching Yao Abstract

It is shown for an

n



n

symmetric positive de nite matrix

T

= (

t

i;j) with negative o -diagonal elements, positive row sums and satisfying certain bounding conditions that its inverse is well approximated, uniformly to order 1

=n

2

;

by a matrix

S

= (

s

i;j)

;

where

s

i;j =



i;j

=t

i;i+ 1

=t::; 

i;j being the Kronecker delta function, and

t::

being the sum of the elements of

T:

An explicit bound on the approximation error is provided.

Gordon Simons Yi-Ching Yao

Dept. of Statistics Inst. of Statistical Science Univ. of North Carolina Academia Sinica

Chapel Hill, NC 27599-3260 Taipei, Taiwan

simons@stat.unc.edu yao@stat.sinica.edu.tw

(2)

Gordon Simons and Yi-Ching Yao 1. INTRODUCTION

We are concerned here with

n



n

symmetric matrices

T

= (

t

i;j) which have negative o -diagonal elements and positive row (and column) sums, i.e.,

t

i;j =

t

j;i

; t

i;j

<

0 for

i

6=

j;

and Xn

k=1

t

i;k

>

0 for

i;j

= 1

;:::;n:

Such matrices must be positive de nite and hence fall into the class of M-matrices. (See, e.g., [2]

for the de nition and properties of M-matrices.) It is convenient to introduce an array 

u

i;j n

i;j=1 of positive numbers de ned in terms of T as follows:

u

i;j =?

t

i;j for

i

6=

j;

and

u

i;i=Xn

k=1

t

i;k

; i;j

= 1

;:::;n:

Then we have

u

i;j

>

0

; u

i;j =

u

j;i

; t

i;j =?

u

i;j for

i

6=

j;

and

t

i;i =Xn

k=1

u

i;k

; i;j

= 1

;:::;n:

(1) Moreover, it is convenient to introduce the notation

m

= mini;j

u

i;j

; M

= maxi;j

u

i;j

; t::

= Xn

i;j=1

t

i;j =Xn

k=1

u

k;k

>

0

;

(2)

k

A

k= maxi;j j

a

i;j j for a general matrix

A

= (

a

i;j)

;

and the

n



n

symmetric positive de nite matrix

S

= (

s

i;j)

;

with

s

i;j =



i;j

t

i;i + 1

t::;

where



i;j denotes the Kronecker delta function.

THEOREM.

k

T

?1?

S

k

C

(

m;M

)

n

2

;

where

C

(

m;M

) = (1 +

M m

)

M

m

2

:

The authors [5] use this theorem while establishing the asymptotic normality of a vector- valued estimator arising in a study of the Bradley-Terry model for paired comparisons. Depending on

n;

which goes to in nity in the asymptotic limit, we need to consider the inverse

T

?1 of a matrix

T

satisfying (1) with

m

and

M

being bounded away from 0 and in nity. Since it is

(3)

Approximating the Inverse of a Matrix

impossible to obtain this inverse explicitly, except for a few special cases, we show that the approximate inverse

S

is a workable substitute, with the attendant errors going to zero at the rate 1

=n

2 as

n

!1

:

Computing and estimating the inverse of a matrix has been extensively studied and described in the literature. See [1], [3], [4] and references therein. In [3], the characterization of inverses of symmetric tridiagonal and block tridiagonal matrices is discussed, which gives rise to stable algorithms for computing their inverses. [1] and [4] derive, among other things, upper and lower bounds for the elements of the inverse of a symmetric positive de nite matrix. In particular, for a symmetric positive de nite matrix

A

= (

a

i;j) of dimension n, the following bounds on the diagonal elements of

A

?1 are given in [1] and [4]:

1 + (

?

a

i;i)2

(

a

i;i?Pn

k=1

a

i;k2) 

?

A

?1i;i 

1 ? (

a

i;i?

)2

(Pnk=1

a

i;k2?

a

i;i)

;

where





n and 0

<





1,



1 and



n being the smallest and largest eigenvalues of

A

, respectively.

The next section contains the proof of the theorem, and some remarks are given in Section 3.

2. PROOF OF THE THEOREM

Note that

T

?1?

S

= (

T

?1?

S

)(

I

n?

TS

) +

S

(

I

n?

TS

)

;

where

I

n is the

n



n

identity matrix. Letting

V

=

I

n?

TS

and

W

=

SV

, we have

T

?1?

S

= (

T

?1?

S

)

V

+

W:

Thus the task is to show that k

F

k

C

(

m;M

)

;

where the matrices

F

=

n

2(

T

?1?

S

) and

G

=

n

2

W

satisfy the recursion

F

=

F V

+

G:

(3)

By the de nitions of

S

,

V

= (

v

i;j) and

W

= (

w

i;j), it follows from (1) and (2) that

v

i;j =



i;j?Xn

k=1

t

i;k

s

k;j

=



i;j?Xn k=1

t

i;k





k;j

t

j;j + 1

t::



=



i;j?

t

i;j

t

j;j ?

u

i;i

t::

= (1?



i;j)

u

i;j

t

j;j ?

u

i;i

t:: ;

(4)

(4)

Gordon Simons and Yi-Ching Yao

and

w

i;j=Xn

k=1

s

i;k

v

k;j=Xn

k=1





i;k

t

i;i + 1

t::



(1?



k;j)

u

k;j

t

j;j ?

u

k;k

t::



=Xn

k=1



i;k

t

i;i



(1?



k;j)

u

k;j

t

j;j ?

u

k;k

t::



+Xn

k=1

t::

1



(1?



k;j)

u

k;j

t

j;j ?

u

k;k

t::



= (1?



i;j)

u

i;j

t

i;i

t

j;j ?

u

i;i

t

i;i

t::

?

u

j;j

t

j;j

t:: :

(5)

Again by (1) and (2), we have

0

< u t

i;ii;j

t

j;j 

M

m

2

n

2

;

0

< u t

i;ii;i

t::



M m

2

n

2

;

so that

j

w

i;jj

a

n

2 andj

w

i;j?

w

i;kj

a

n

2 for

i;j;k

= 1

;:::;n;

where

a

= 2

M=m

2. Equivalently, in terms of the elements of

G

= (

g

i;j):

j

g

i;jj 

a

and j

g

i;j?

g

i;kj 

a; i;j;k

= 1

;:::;n:

(6) We now turn our attention to (3), expressed in terms of the matrix elements

f

i;j and

g

i;j in

F

and

G;

respectively, and the formula for

v

i;j in (4):

f

i;j =Xn

k=1

f

i;k(1?



k;j)

u

k;j

t

j;j ? n

X

k=1

f

i;k

u

k;k

t::

+

g

i;j

; i;j

= 1

;:::;n:

(7) The task is to show j

f

i;j j

C

(

m;M

) for all

i

and

j:

Two things are readily apparent in (7). To begin with, apart from the factor (1?



k;j) in the rst sum, which equals one except when

k

=

j;

the rst and second sums are weighted averages of

f

i;k

; k

= 1

;:::;n

; the positive weights utk;jj;j and ut::k;k each add to unity in the index k. Secondly, the index

i

plays no essential role in the relationship; it can be viewed as xed. If we take

i

to be xed and notationally suppress it in (7), then (7) assumes the form of

n

linear equations in the

n

unknowns

f

1

;:::;f

n:

f

j =Xn

k=1

f

k(1?



k;j)

u

k;j

t

j;j ? n

X

k=1

f

k

u

k;k

t::

+

g

j

; j

= 1

;:::;n:

(8) Instead of solving these equations, we will show that under the bounding conditions

g

j

a; g

j

g

k

a; j;k

= 1

;:::;n;

(5)

Approximating the Inverse of a Matrix

(see (6)) any solution of (8) must satisfy the inequalities

j

f

j j 1 2

1 +

M m



a; j

= 1

;:::;n;

(9)

so that j

f

j j

C

(

m;M

)

; j

= 1

;:::;n;

thereby completing the proof.

Let

and

be such that

f

= max1kn

f

k and

f

= min1kn

f

k

:

With no loss of generality, assume

f

j

f

j

:

(Otherwise, we may reverse the signs of the

f

k's and proceed analogously.) There are two cases to consider:

Case I: f

0

:

Then

f

=Xn

k=1

f

k(1?



k; )

u

k;

t

; ? n

X

k=1

f

k

u

k;k

t::

+

g



n

X

k=1

f

k

u

k;

t

; ? n

X

k=1

f

k

u

k;k

t::

+

g

=Xn

k=1

f

k



u

k;

t

; ?

u

k;k

t::

+

g



f

X k2A



u

k;

t

; ?

u

k;k

t::

+

g

;

where

A

=

k

:

u

k;

=t

;

> u

k;k

=t::

:

Let



denote the cardinality of

A;

and observe that

X

k2A



u

k;

t

; ?

u

k;k

t::





M

+

M m

(

n

?



) ?

m

m

+

M

(

n

?



) 

M

?

m

M

+

m;

(10)

the rst inequality being an immediate consequence of the constraints

m



u

i;j 

M

(see (2)) and the sum formulas in (1) and (2), the second inequality taking into account that the middle expression in (10) is a concave function of



(when viewed as a continuous variable between 0 and

n

), with its maximum occurring at



=

n=

2

:

Thus,

f



f

X k2A



u

k;

t

; ?

u

k;k

t::

+

g



f

M

?

m

M

+

m

+

g



f

M

?

m M

+

m

+

a;

so that

f

 1 2

1 +

M m



a

=

C

(

m;M

)

;

thereby establishing (9) and completing the proof.

(6)

Gordon Simons and Yi-Ching Yao Case II: f

<

0

:

Let

h

k =

f

k?

f

0

; k

= 1

;:::;n:

Then

h

=

f

?

f



n

X

k=1

f

k

u

k;

t

; ? n

X

k=1

f

k

u

k;

t

; +

g

?

g

= Xn

k=1

h

k

u

k;

t

; ? n

X

k=1

h

k

u

k;

t

; +

g

?

g

= Xn

k=1

h

k



u

k;

t

; ?

u

k;

t

;

+

g

?

g



h

X k2A



u

k;

t

; ?

u

k;

t

;

+

g

?

g

;

where

A

=

k

:

u

k;

=t

;

> u

k;

=t

;

:

The argument from this point proceeds analogously to that for Case I. Letting



denote the cardinality of

A;

one obtains

X

k2A



u

k;

t

; ?

u

k;

t

;





M

+

M m

(

n

?



) ?

m

m

+

M

(

n

?



) 

M

?

m M

+

m;

which leads to

h



h

M

?

m

M

+

m

+

g

?

g



h

M

?

m M

+

m

+

a;

so that

f



h

 1 2

1 +

M m



a;

thereby establishing (9) and completing the proof.

3. REMARKS

While our proof of the theorem is somewhat long, we do not see how to simplify it by using any of the well-known properties of M-matrices.

The bound C(m;Mn2 ) on the approximation error is a product of two factors, one depending on

m

and

M

, the other on

n:

For large

n;

with

m

and

M

held bounded away from 0 and in nity, the elements of

S

(and hence of

T

?1) are all of order 1

=n

, and the errors (i.e., the elements of

T

?1?

S

) are uniformly

O

(1

=n

2) as

n

!1. This fact is crucially used in the work of [5].

A particular case of the matrix

T

, described below, shows that the factor 1

=n

2 is best possible in the sense that any bound of the for

C

e(

m;M

)

=

(

n

) requires

(

n

) =

O

(

n

2) as

n

!1; no faster growth rate than

n

2 is allowed. On the other hand, it is natural to ask whether the factor

C

(

m;M

) is best possible. To clarify the issue, for given integer

n

and given

m

and

(7)

Approximating the Inverse of a Matrix

M;

(0

< m



M <

1)

;

let

Q

n(

m;M

) denote the set of

n



n

symmetric positive de nite matrices satisfying (1) with

m



u

i;j 

M; i;j

= 1

;:::;n;

and de ne

C

o(

m;M

) = supf

n

2 k

T

?1?

S

k:

T

2

Q

n(

m;M

)

; n

= 1

;

2

;:::

g

:

It follows from the theorem that

C

o(

m;M

)

C

(

m;M

) = (1+Mm) mM2

:

But for the special matrix

T

satisfying (1) with

u

1;1 =

M

and

u

i;j =

m

for all other (

i;j

)

;

we nd that

(

T

?1)i;j =

8

>

>

>

>

<

>

>

>

>

: 2

2M+(n?1)m for

i

=

j

= 1

;

1

2M+(n?1)m for

i

= 1

;j

6= 1 or

i

6= 1

;j

= 1

;

3M+(2n?1)m

(n+1)m(2M+(n? 1)m) for

i

=

j

6= 1

;

M+nm

(n+1)m(2M+(n? 1)m) for 1 6=

i

6=

j

6= 1

:

So (

T

?1?

S

)1;1 = (M+(n?1)m?2)(2MM+(n? 1)m)

;

from which it follows that

C

o(

m;M

)  2mM2

:

The same matrix

T

justi es the constraint on

(

n

) described above.

The gap between 2mM2 and (1+Mm)mM2 suggests that there might be room for improvement in our bound. Indeed, by computer, we have numerically inverted a very large number of matrices of various dimensions (some as large as 300300) and found that the inequality

n

2 k

T

?1?

S

k 2mM2 holds in all cases. It would therefore be interesting to see whether

C

o(

m;M

) = 2mM2

:

We nish with one nal observation. Surprisingly, it is possible to evaluate the second sum

in (7) explicitly: n

X

k=1

f

i;k

u

k;k

t::

=?

n

2

u

i;i

t

i;i

t::;

which is identical to, and permits a cancellation with, one of the three terms de ning

g

i;j (cf., (5)). To obtain this, one multiplies both sides of (7) by

t

j;j

;

adds over

j

(

j

= 1

;:::;n

)

;

and carries out the suggested algebra. While we have not found much use for this identity, it does show that

f

;

appearing in the proof of the theorem, is strictly negative. Since, as it turns out,

f

can be positive or negative, neither case described in the proof is super uous.

REFERENCES

[1] G. H. Golub and Z. Strakos, Estimates in quadratic formulas, Numerical Algorithms,

8

(1994), 241-268.

[2] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (1991), Cambridge University Press, New York.

[3] G. Meurant, A review of the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl.,

13

(1992), 707-728.

(8)

Gordon Simons and Yi-Ching Yao

[4] P. D. Robinson and A. J. Wathen, Variational bounds on the entries of the inverse of a matrix, IMA Journal of Numerical Anal.,

12

(1992), 463-486.

[5] G. Simons and Y-C Yao, A large sample study of the Bradley-Terry model for paired com- parisons, in preparation.

References

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