Minimizing a norm of a matrix

LINEAR

ALGEBRA

Minimizing

AND

a Norm

ITS

447

APPLICATIONS

of a Matrix

J. J. A. MOORS Central

of Statistics

Bureau

The Hague,

The

Communicated

Netherlands

by Alan

J. Hoffman

ABSTRACT

In their article, Abel and Waugh norms of matrices the following

C = (l/k)A

[l] have mentioned

is less than unity:

I. COMPUTATION

number,

In

is derived for the value of k minimizing

under the condition

that the minimum

this is the most important

the

value of N(C)

case.

OF AP

AP, where

To compute

the problem of minimizing

I, where k is a scalar and A a given matrix.

note a general formula

norm N(C) = max,Z&jl

negative

-

A is a given

the following

k(1 + C) with k a scalar;

method

matrix

and fi a fractional

is proposed

from the binomial

in

theorem

[l].

Let

or A =

it follows that

r+pc+?-c,,+.... 2!

The sum of the first several approximation

terms of this power series can be used as an

to A@. Convergence

(if any) is fastest

when the norm of

C is minimum. Many norms of matrices a general

formula

of C is smallest.

are possible, however, and not for each of them

is known For

that

gives the value of h for which the norm

the norm N(C) = max x t j

we present

such

a general

formula,

lcijj

(2)

provided

min N(C) < 1.

(3)

k

Linear Copyright

0

1969

Algebra

and

by American

Its Applications Elsevier

Publishing

2(1969),

447-449

Company,

Inc.

44s

J. J. A.

This condition, neously 2. THE

however, imposes no serious restriction

ensures

the convergence

IMPLICATIOiYS

Let

A = (Q),

because it simulta-

of (1).

OF COXDITIOK

We now investigate

MOORS

(3)

the implications

of condition

(cpj) = C = (l/k)A -

(3).

I, and hence for

i-j

C-1) for If condition

(3) holds,

i#-i

,ci,/ < 1 holds for all i and i and a certain

of k, and in particular

laii/k -

l/ < 1 for all i.

0 < aii/k < 2 holds for all i, and therefore So we may

aii.

changing

value

the inequality

k has the same sign as all

suppose for all i;

aTi > 0 because

Hence

k>O

all these signs does not change

(5)

the value of N(C).

We

now write for all i,

(6)

(7) where fi(h) is defined

only for k > 0.

We find immediately if

if

:

a, >, a,,,

then

f<(k) > 1,

a, < a,,,

then

ii(K) =

and fi(k) reaches It follows

that

X(C)

Algebra

and

Its

k < aii

for

k >, aj,

for k = aij.

< 1 if and only if ai < aii

Lineav

its minimum

for

A@%xztiuns

for all i.

rS(1969),

447-449

(8)

MINIhUZING

A NORM

Condition

OF

449

A MATRIX

(3) thus leads to the inequality

(8), which we shall suppose

to hold in the following. 3. THE

BRANCHES

OF THE

We want to examine of the point

JiR(k)

1,

k

are increasing

all values

of the graph of ji(k) on both sides

and to this

hi + ai) _

=

z

OF f,(k)

the branches

k = aii separately, f,yk)

These

GRAPH

functions

= +j

+

of (aii + a,), (ai -

a,,),

for k > 0:

1.

respectively,

for

of k ; hence max fiL(k) = -i max (a,, + ai) i / max fiR(k) = 1 -

-k mm (a,, -

‘

L

Besides,

end we define

max(fiL(k),

1 for all k.

(10)

ai)

fiR(k)) = fi(k) ; therefore

min N(C) = min (max fi(k)} = min [max {max fiL(k), max f,‘(k)}], k

k

which (10).

s

value is given

k

by the point

i

of intersection

This follows from the monotone

respect

to k.

The resulting

reaches

its minimum,

z

formula

character

of the two functions of these functions

with

for the value of k, for which N(C)

is thus

k = maxi(aii

+ ai) + min,(aii

-

ai)

2 Applying maxJa,, found

this result

to the numerical

+ ai) = 1.0 and mini(aii by the authors

-

example

given in [l]

we obtain:

ai) = 0.2, so that k = 0.6, the value

by inspection.

REFERENCE 1 F. V. Waugh Assoc.

Received

62(Sept.

July

and M. E. Abel,

On fractional

powers of a matrix,

J. Am.

Statist.

1967).

23,

1968 Lineav

Algebra

and

Its

Applications

2(1969),

447-449