On classes of copositive matrices

LINEAR

On R.

ALGEBRA

Classes W.

ITS

295

APPLICATIONS

of Copositive

Matrices

COTTLE*

Stanford G.

AND

J.

University,

Palo Alto,

California

HABETLERt

C. E.

LEMKE:

Polytechnic

Rensselaer

Communicated

by

Institute,

Alan

Tmy,

New

York

J. Hoffman

ABSTRACT Characterizations matrices,

are given

and their quadratic

semidefinitc

of copositive,

forms,

strictly

together

copositive,

and copositive

with relationships

plus

of these with positive

matrices.

1. INTRODUCTION

Throughout n whose

this paper, M denotes

associated

quadratic

a real symmetric

form we denote

matrix

of order

by

Q(z) = z’Mz. Prime,

‘, denotes

transposes * Research Contract

matric

of columns, partly

AT(04-3)

transposition;

rows

such as z’ in (1).

supported

326

PA

under

18, and

(1) The symbol

contracts: Office

of

are usually

U.S. Naval

Atomic

A,,j

written denotes

Energy

Research

as the

Commission

Contract

ONR-N-

00014-67-A-0112-0011. t Grant

Research

partially

supported

by

the

National

Science

Foundation

Research

NSF-GP-7958.

r Research

partly

supported

by the National

Science

Research

Grant

NSF-GP-

7960.

Linear Algebra Copyright

0

1970

by

American

and Its Applications Elsevier

Publishing

3(1970),

295-310

Company,

Inc.

296

R. W.

row i, column

COTTLE,

j component

subclasses

of copositive

definitions

and notations. 1.1.

DEFINITION

G.

A.

of the matrix

matrices,

for which

AND

If and only if

copositive

MEC,

z2 0

implies

strictly

MEC”,

220,

z#O

MEG+,

MEC;

copositive

plus

MEP”,

z# 0

positive

semidefinite

MEP+,

Q(z) 3 0,

to provide

between

a complete

Q(z)>O,

and Q(z)=0

implies

Q(z) > 0,

all

z.

these classes were developed

characterization

results of the present note, although results of other investigators

Q(z) 3 0,

Mz = 0,

implies definite

In [5], various relationships

with

the following

implies

220,

positive

an attempt

LEMKE

above

Denoted

copositive

C. E.

We are concerned we provide

M and Q as described

With

M is

J. HABETLER,

of independent

and may be considered

of the class Cf.

in The

interest, tie in with extensions

of those

in [5], some of which we shall recall as they pertain to the present study. To begin with, we have the following P* c c

strict inclusions:

P+ c

c*c

C’CC.

Thus the diagram depicts relations among classes of copositive In addition,

we have at once two important

properties,

namely,

under principal rearrangements and the property of inheritance. rearrangement

matrices. closure

A principal

$f of M is a matrix obtained from M by an equal permuta-

tion of rows and columns: M = P’MP, where P is a permutation of the above classes.

Let T, momentarily,

any principal submatrix

of M, A E T if M E T;

the latter is, by definition,

(Thus we mean the classes to contain matrices

The notion of inheritance

suggests the following

used in the sequel. Linear

Algebra

denote any one

Then Ii? E T if and only if M E T and, if A denotes

the property of inheritance. of all orders.)

matrix.

(2)

and Its Applications

3(1970), 295-310

definition,

ON

CLASSES

OF

COPOSITIVE

DEFINITION 1.2.

Let

let M have order n. every

principal

Thus,

submatrix

Theorem

desire

a more

complete

of C*.

of Theorem

implies

to expand

characterization

(See

‘M is T of order s’,

class P+ and perhaps and proof.

It thus

of C+ would require

a

of Cm and of C are given in by practical

and game theory

some previously

and

if and only if

M E T.

in the statement

Our work has been stimulated programming

n,

8 in [5], giving an improved

Characterizations

15, 10, 91, on the other. 2.

7'

7 <

is in T.

7

The roles of the familiar

class C* are emphasized

that 3.

‘M is T of order

of Cf.

characterization Section

of M of order

2.1 is a sharpening

less familiar

quadratic

a fixed one of the above classes,

and ‘M is T of order n’ means

characterization appears

T denote

We say ‘M is T of order r’,

for example,

for s < Y;

297

MATRICES

considerations

in

[12] on the one hand, and the

published

also [2],

[3],

determinantal and

criteria

[13,

[B].)

COPOSITIVE PLUS MATRICES THEOREM 2.1.

principal

Let M have order n.

rearrangement

M E C+ if and only if there is a

A? of M which in block form is

(3) such that (i) (ii) (iii) Before together

AEPf,

A is of order 7, O
B = AB, D -

for some B;

B’AB

presenting

E C* (hence DE C*). the proof, we recall a definition

with an associated

DEFINITION 2.1. in the null space

characterization

A column

in [5],

C+ and P+.

Z is a flat point of M if and only if ,? is

of M: Mf=O.

From the standpoint also the gradient

introduced

of the classes

of analysis,

(4)

a flat point of M is a zero of Q at which

of Q vanishes. Linear

Algebra

and

Its

Applications

3(1970), 295-310

R. W.

298

As an immediate

COTTLE,

G.

consequence

J. HABETLER,

of the definition,

of M, we see from the form of Q that,

C. E.

LEMKE

and the symmetry

if 2 is a flat point of M,

Qb + 4 = Q(z), In Theorem

AND

for all .z.

5 of [5], the following

(5)

characterizations

were observed

for M#O: (i)

ME P+ if and only if every

Q(z) is positive (ii)

zero of Q is a flat point of M and

for some z;

M E C+ if and only if every nonnegative

of M and Q(z) is positive In particular,

the class N of nonnegative

only if M,,j 3 0, all i and i) is a subclass the strictly nonnegative

class N * by MEN*

matrices

yc, positive guished

components,

z1 + 2

of positive

From

so that,

components,

nonnegative

flat

=

n.

Let that

must

with more than

points

must

have

the

First

is in P+. suppose M E C+.

There

Z > 0 and Mi = 0 implies

r,, = 0 if and only if ME C*, and Y of the theorem

0 > 0 large enough

flatpoints

the Y,, components

0. We further define s,, as the order of the largest

Then

r,=O.

y.

the set

in fact, along with M, we have ~a distin-

of M which

Proof of Theorem 2.1.

Case 2.

components

(4), we see that

would be a flat point

such that

r,, components submatrix

Case 1.

of C+.

if zi and z2 are two nonnegative

r0 positive

otherwise

components

other n principal

Further,

with just

be the same;

Finally,

if and only if

flat points of M is a (closed) convex cone, and consequent-

ly ~a is well-defined. of M, both

and

1, if M has order n, we

number

flat point of M may have.

of all nonnegative

by MEN+

to Theorem

by ra (0 < r0 < n) the largest

Defining

of C*, hence of Cc.

M,,? = 0, all j, is a subclass

For the proof of and corollaries a nonnegative

M (M E N if and

if and only if MEN

the class N+ of flatly nonnegative M, defined ME N and Mi,i = 0 implies

z.

of C, but not of C+.

M,,i > 0, all i, we have that N* is a subclass

denote

zero of Q is a flat point

for some nonnegative

Z > 0 be a flat point

are three cases. i = 0.

Hence

may be taken to be 0. of M.

For any z, and

z + OZ >, 0:

Q(z)= Q(z+ fW > 0 by (5). Linear

Hence Algebra

M E P+, and Y of the theorem

and Its Applications

3(1970), 295-310

(6) may be taken

to be n.

ON

CLASSES

OF

COPOSITIVE

299

0 < y. < n. Let P’ be a permutation

Case 3.

the distinguished columns

MATRICES

r, components

to the top:

x and y, (x, y> means

so that

P’z = (x, y) ; Correspondingly,

; ii

the principal

matrix

which permutes

z = (x, y)

(generally,

for

).

P’Z = (2, 0), rearrangement

where

x >o.

(‘1

is

and

z’Mz = (P’z)‘(P’MP)(P’z)

= (x, y) ‘&?(i(x,y),

(9)

so that Q(z) = Q(x, y) = x’Ax + 2x’By

+ _y’Dy.

(10)

x > 0.

(11)

In particular, 0 = Q(a) = Q(s, 0) = f’A5, By inheritance,

A EC+, hence Ai! = 0.

of A, so that,

by Case 2, A E Pi,

X is thus a positive

which is assertion

flat point

(i) of the theorem,

with r = yO. With reference

we readily sequence

to A, conversely,

deduce that

of which is that

for 0 satisfying

if x0 is a nonnegative

(x0, 0) is a nonnegative B’x”

= 0.

flat point of A,

flat point of ai,

In fact, if x is any flat point of A,

x0 = x + 82 > 0, we have

that

0 = B’xO = B’x + OB’x = B’x. We conclude that

that

this implies

a con-

(12)

B’x = 0 for all x such that Ax = 0. It is standard that B = AD, for some B, which is assertion (ii).

In terms of this B, (10) may be written Q(x, y) = (x + By)‘A(x

+ By) + y’(D -

Now, for any x and 8 for which

x + OX 3

Q(x>Y) = Q(x + 0% Linear

Algebra

B’AB)y.

(13)

0, we have

~1,

and Its Af$lications

(14) 3(1970), 295-310

K. W. COTTLE,

300

G. J. HABETLER,

AND

Now, for given

by using (5), so that, for y 3 0, we have Q(x, y) > 0. y > 0, take x = -

By,

so that, from

(15)

But,infact,ify>O,y#Oancly’(D-PAB)y=O,

then for 0 large enough negative

(13),

fS’AB)y = Q(x, y) > 0

y’(D andD--i?ABEC.

C. E. LEMKE

flat

point

contradiction.

and x = -

of M

with

This proves

By,

more

assertion

(x + 02, y)

than

would

Y, positive

be a non-

components,

a

(iii).

Next, suppose that (i), (ii), and (iii) hold for some principal rearrangement &‘ of M. both

We show that M E C+.

both terms must vanish and, in particular y = 0. Then we would have x’Ax (x, 0) is a nonnegative COROLLARY

and, if Q(z) = 0,

by (iii), this would imply that

= 0 and, as in well known, this implies

by (ii), Ax = 0 implies

Finally,

Ax = 0, by (i).

We note first that, for z > 0,

so that ME C;

terms of (13) are nonnegative,

flat point of M;

B’x = 0, hence z =

the proof is now complete.

2.1. If M E C+ and ii% is as in the assertion of Theorem 1,

then, for any y > 0,

M(Y) Proof.

=

P+. (xi, yzy)e

For x and nonnegative

scalar 8 we may write

(16) But also

so that M(y)EP+.

O.Q(-x,.)=Q(~.-ey)=(_~~M(y)j_~)‘,

COROLLARY Proof. matrix

2.2. If ME C+ and r,, < n, then s,, > y0 + 1.

In Corollary

of order n -

r,.

order r0 + 1, completing We terminate

2.1, take y = ei, the ith column Then M(ei)

is a principal

of the identity

submatrix

of M of

the proof.

this section with an example

of a class of symmetric

matrices which are in C+ if and only if they are in Pf ; namely, the class Linear

Algebra

and Its Applications

3(1970), 295-310

ON CLASSES

of matrices

OF COPOSITIVE

M for which M,,j

of Hoffman, Let

clear that

= & 1 and Mii = 1. The results are those

and are reported

et denote

a column,

in [ll,

11.

each of whose t components

all such matrices

only such matrix

301

MATRICES

is + 1.

M of order 2 are C+, and, in fact,

It is The

P+.

of order 3 which is not in C is

c 1

-1 -1

A= (e3’Ae3 is negative). M EC

THEOREM 2.2.

if

-1

-1

-1 1

-1 1

1

and only if no $vincipal

submatrix of M has

the form (17). In this class are the matrices

of rank

1 of the form

zi=(__ :1-J_ ::_$ (two diagonal blocks of + l’s; these

are clearly

THEOREM

labeled

have an inductive

of C and C*.

Adj M,

It should be observed

that

of M, which,

DEFINITION leas positive

The matrix

and Det M denotes

determinant, 3.1.

and

rearrangement

of

(Hence M E C+ if and only if M is in P’.)

(Adj M)M

adjoint,

Being of rank 1,

STRICTLY COPOSITIVE MATRICES

theorems

the symmetry

l’s).

M E C+ if and only if a principal

2.3.

3. COPOSITIVE AND

The next

-

in P+.

M has the form (18).

acterizations

other components

(18)

= M(Adj

of a matrix

however,

z will

M is

of M, so that

M) = (Det M)I.

all arguments

A column

and mean to give char-

the determinant

and eigenvalue.

negative

flavor

of cofactors

(19)

made only on Q do not invoke

is prominent First,

in considerations

of

some preliminaries.

be called posineg

if and only

if z

components. Linear

Algebra

and Its

Applications

3(1970),

295410

K.

302

W.

COTTLE,

Observe the trichotomy or z is posineg. LEMMA

positive

3.1.

THEOREM

J. HABETLEK,

LEMKE

of the following

lemma

is omitted.

that

Suppose that M has order n, and that M is C of order

3.1.

if and on&

if

Adj M > 0,

(9

DetM

(ii)

of minimum

and


(in which case, M is Pi of order n -

Proof.

C. E.

If i > 0 and ZEis a posineg column, there exist unique

Then M$C

eigenvalue

rlND

that, for each z in real n-space, z > 0, z ,( 0,

The proof

scalars O1 and 8, such

n--l.

G.

magnitude

1,

M-l

< 0, and M has one negative

and it has a positive

First, suppose that M $ C.

eigenvector).

Then, for some 2 > 0, Q(Z) < 0.

Then Z > 0, since M is C of order n - 1. If z0 is posineg, then so is Gl = KZO- 2 for K large enough. The line joining KZOand i is the set of points of the form z = 2 + 66, for scalar 8; furthermore, in 0 on this line.

Consider z1 and z2, the nonnegative

Q(z”) > 0, since M is C of order n -

3.1.

some point on the segment hence Q is positive z1 and z2.

joining

It follows that Q is zero at

f and z1 and that joining

on the line except

In particular,

1.

(possibly)

Q(z”) = (~/K~)Q(KzO) > 0.

joining

That is, at posineg

In particular,

component.

z” 3 0, z” < 0, or z” is posineg.

Q(Z), Q(z’J) > 0 in any case;

2 and za;

on the segment

points, Q is positive. Either,

Q is quadratic

columns of Lemma

consider points z” which have a zero Since Q(-

that is, M is P+ of order n -

z) =

1.

It follows that, for each i, (Adj M),,i > 0, since these are the determinants

of the principal

submatrices

of order n -

1.

Now, since Q is negative for some z, M has a negative eigenvalue, it +r < 0.

call

Let zr # 0 satisfy

Mzl = &z’ ;

so that

zl’zl = 1

Q(zl) = $r < 0.

(21)

But Q(z) < 0 only if z > 0 or z < 0. It follows that we may take z1 > 0. Let the columns Linear

Algebra

of the orthogonal

and Its

Applications

matrix

3(1970),

295-310

ON

CLASSES

OF COPOSITIVE

MATRICES

303

2 = (21,22,. . ) z”) comprise

a full set of eigenvectors

of M.

,$‘zi zz 0

Then

i=2,3

the condition

,. . .,n

(22)

asserts that ,z2, z3,. . ., zn are posineg, so that Q(#) > 0, i = 2, 3,. . . , n. Now the flat of all points satisfying z’zl = 0 is, on the one hand, composed of z = 0 and posineg linear

points,

and, on the other

hand, is the subspace

of

combinations

z = p2z2 + p3z3 + . . . + pnP, and, if $i denote

the eigenvalues

(23)

of M, since Q(z) > 0 for all z # 0 on

the flat,

Q(z) = from which

~2~+2

we conclude

+

~3~$3

+

. . . +

pn2+n

>

(24)

0,

that

i=2,3,...,n;

#i > 0,

(25)

and hence Det M < 0, since

Det M is the product

To see that

M-l

(26)

of the $i.

< 0, let a > 0 and define the simplex S(a) = {z >, 0: z’a = l}.

The minimum

Q on this compact

Q < 0 since MT

is taken

the Lagrangian

on in the relative

condition Li’fi? =

and Q =

KZ’U

<

0 implies

K <

0.

Clearly,

1, it follows that Z > 0; interior

of S(a).

Hence,

holds, KU;

(28)

It follows that,

M-la In particular,

on at some z 3 0.

C and, since M is C of order n -

that is, the minimum necessarily,

set is taken

(27)

for all a > 0,

< 0.

(29)

for a = e + Oei and 0 > 0,

M-b and, if some component

+ BM-W

< 0;

of the ith column, Linear

Algebra

and

M-lei,

(30) were positive,

Its Applications

3(1970),

then, 295-310

304

R. W.

COTTLE,

for 6 large, the component contradicting

(30).

Hence

G.

J. HABETLER,

AND

C. E. LEMKE

of the left side of (30) would M-l

< 0;

be positive,

and

Adj M 3 0. It should be noted that the nonnegative

(31)

matrix -

for otherwise it follows from the symmetry (M is C of order n classical theorem of the theorem Conversely,

1 but not copositive)

of Frobenius

that M E C.

dicting

the

assertions

are valid. if ME C, the assumption,

0.

From (i), z* 3 0. This completes

M E C.

The matrix matrix

Therefore

[7] applies and the remaining for contradiction,

and (ii) hold, implies that Adj M is nonsingular; (Adj M)b f

> 0 is irreducible,

M-l

of M and the other hypotheses

that both (i)

hence, for b > 0, z* =

But Q(z*) = (Det M)z*‘b

< 0, contra-

the proof.

A in (17) illustrates

the theorem.

This noncopositive

A has eigenpairs $I=

yet is C (indeed

-

zi = (1, 1, l),

1,

$2 = 2,

22 = (2, -

&=2,

23 = (0, 1, -

K

+

l),

l),

P+) of order 2.

More generally, consider A (K) = A + 4, = 43 =

1, -

2.

KI

with eigenvalues & =

K

1,

-

have

We

and

Det

A(K)

=

(K

-

l)(~ + 2)“. (32)

A(K) illustrates

Theorem

3.2, to follow,

for 0 <

K

Theorem 3.1 does not hold if M is not symmetric, 0 o

M=

i shows.

It does hold, of course,

0

3(1970),

1.

1

for the symmetric

+(M+M’)=(_; Linear Algebra and Its Applications

-1

<

as the simple example

-a). 295-310

part

ON CLASSES However,

OF COPOSITIVE

305

MATRICES

we have the following result (in which the symmetry

is temporarily

assumption

waived).

COROLLARY

3.1.

If M is

of

C

order n -

(9

Adj M 3 0,

(ii)

Det M < 0,

1 awd

then M q! C, and (iii)

Proof. of Theorem

Adj(M

+ M’) 3 0,

Det(M

+ M’) < 0.

If (i) and (ii) hold, M $ C, since the converse 3.1 does not use symmetry.

since the quadratic COROLLARY

forms associated

Assertions

part of the proof

(iii) and (iv) then follow

with M and $(M + M’) are identical.

3.2. Let M be of order n, and suppose M is PC of order

1. Then M is not P+ if and only if Det M < 0, in which case there

n -

exists a real matrix F of order n with F2 = I such that FM-IF Proof.

The first part is standard

P+ yet is Pi associated

of order n -

eigenvector

and requires

1, it has a negative

z all of whose components

let F = I. Otherwise there exists a (sign-changing) type

such that

Fz > 0.

The matrix

FMF

Theorem

eigenvalue are nonzero.

If M is not ;I with

an

If z > 0,

matrix F of the required

Now we have

iiz’z = z’Mz = (z’F)(FMF)(Fz)

hence

no proof.

< 0.

is clearly

Pi

of order n -

< 0. 1 but not copositive

and

3.1 applies : Adj FMF

> 0,

Det FMF

(: 0,

whence FM-IF The next theorem

= (FMF)-l

< 0.

is similar but, because

it deals with C*, it is more

involved. Linear

Algebra

and

Its

Applicatimzs

3(1970). 295-310

306

R. W.

G.

J. HABETLER,

AND

C. E.

LEMKE

SzL$$ose that M has order n, and that M is C* of order

THEOREM 3.2. n -

COTTLE,

Then M g! C* if and only if

1.

Adj M > 0,

0)

and

DetM
(ii)

(in which case, M is P* of order n -

1, so that Rank M > n -

1). Further-

more, if M 4 C*, then either (a)

which is true if and only if Det M < 0 (in which case M

M 0 C;

has exactly

one negative eigenvalue

it has a positive

eigenvector),

M E C;

(b) ME Pi,

of strictly minimam

magnitude,

and

07

which is true if and only if Det M = 0 (in which case

Rank M = n -

1, and the eigenvector associated with eigenvalue

0 is positive). Proof.

First let us assume that M $ C*.

Case 1.

M 4 C. Exactly

There are two cases.

as in the proof of Theorem 3.1, we conclude

that for posineg z, Q(z) > 0.

But, if z has a zero component,

either z is

posineg or z >, 0 or z < 0.

In the latter two cases, Q(z) = Q(-

since M is C* of order n -

1.

order

n -

1;

and Q(z) < 0 implies

either

z > 0 or z < 0.

(Adj M),,i > 0, for all i, since these are the (positive) principal

submatrices

A’ote.

of M of order n -

The argument of M.

the symmetry not necessarily

thus far, involving matrix

Further

determinants

of

1. only Q, does not require

That the principal submatrices

symmetric,

z) > 0,

it follows that M is P* of

In particular,

are positive

of a positive definite,

is well known;

a proof

is given in [5] .) Exactly single

eigenvalues holds.

as in the proof of Theorem

negative

eigenvalue

are positive,

with

that M has a

3.1, we conclude

positive

eigenvector,

so that (26) holds.

and

that

In the same manner,

other (31)

But, if, for some i, z = (Adj M)ei I+ 0, we obtain Mz = (Det M)ei

by using (19), and therefore Q(z) = (Det M)z’ei = (Det M)(Adj

Adj M > 0.

which

can only hold if t > 0.

Hence,

Linear

Algebra

3(1970), 295-310

and

Its A@lzcations

M),,i < 0,

(33)

ON

CLASSES

OF

307

COPOSITIVE

MATRICES

assertions

of the theorem

The remaining

follow from Perron’s

theorem

[71. M E C.

Case 2. is taken

on for some Z > 61 But then

M has a positive 2.1),

Now Q, the minimum

flat point 2.

Then

of Q over S(a) in (27) is 0 and

(28) implies that

K = 0.

That

is,

(by Case 2, in the proof of Theorem

ME P’.

To show that M is P* of order n component. n -

1.

1, let z0 be a column with a zero

If z” > 0 or z” < 0, then Q(zO) > 0, since M is C* of order

Otherwise

z” is posineg

and, for some specific

.? + Bz” > 0 but has a zero component.

Then,

Q(BzO) = /YQ(z”), or Q(z”) > 0, and M is P* Det M = 0, and Rank M = vz 0).

In fact, Rank

positive

0, z* =

using (5), 0 < Q(z*) = of order

n -

Hence

1.

1, since Adj M # 0 (because (Adj LU)~,~>

(Adj M) = 1, and its symmetry,

together

with MZ = 0,

implies Adj M = ~2 from which,

since

It remains

Z > 0, follows

(34)

Adj M > 0.

1 and (ii) cannot if ME C*, (‘)

to show that,

jointly

hold;

for, if so, with z = (Adj M)ei, Q(z) < 0, as in (33), which implies M $ C*. This

completes

Again,

the proof.

the nonsymmetric

matrices

Z4=c_i

-:)

with

DetA=O

and with illustrate

that

the theorem

4. DETERMINANTAL

Det R > 0

is not valid

for nonsymmetric

CRITERIA

In [5] we briefly reviewed the determinantal by Motzkin sufficient, misquoted

[13;

see also lo]

and the criterion by Hall

M

copositivity

which is known

test suggested

to be necessary

but not

of Garsia which, it appears, was inadvertently

[9, p. 2741. Linear

We are indebted Algebra

and

Its

to Professor

Applications

Garsia for

3(1970),

295-310

R. W. COTTLE, G. J. HABETLER,

308 showing

us his unpublished

Baumert

correct statement

[8j on this subject,

communication

the work in the preceding

drew our attention

strict copositivity*

to Professor

Motzkin’s

that they may be derived THEOREMS4.1.

(Garsia

[S], Baumert 1. Set M(K)

Then M is not copositive

negative.

determinantal

=

K the quantities

Let M have order n and

[4]). M

KI, and D(K) = Det M(K),

+

denote the cofactors of the last YOW

if and only if for sufficiently

E,(K)D(K),

E,(K)D(K),

. . , E,(K)D(K)

the quantities E,(O)D(O), E,(O)D(O), . . , E,(O)D(O)

THEOREM 4.2.

(Motzkin

submatrix

is C* of order n -

for M $ C, M(K)

$ C for K arbitrarily

quantities Ei(~)D(~) (with components which precludes Next,

If M is C of order n 1. For K positive,

Since the Ei(~)

D(K) < 0.

(This means to

of the diagonal components.)

Proof of Theorem 4.1. M(K)

is strictly

for which the cofactors

of the last YOWare all positive has a positive determinant. include the positivity

all

aye all negative.

A symmetric matrix

[5, p. 2381).

if and only if each princi$al

small are

if and only if

In the event that E,(O) # 0, M is not copositive

copositive

test of we shall

criteria, and take the liberty of observing

where K > 0. Let E,(K), E&K), . . . , E,(K) of

sections, Dr. Baumert

from our theorems.

M be copositive of ordern -

values

us of his own

test of copositivity.

[14; 15, p. 2381. For the sake of completeness,

state these revised determinantal

of M(K).

and to Dr. L. D.

[4], apprised

and proof of Garsia’s determinantal

After we had completed kindly

notes

who, in a private

AND C. E. LEMKE

comprise

are all negative.

Ei(~)),

z(K)‘&~(K)z(K)

1, then, by inheritance,

Theorem

small.

3.2 applies, in that,

Hence

the last column Conversely, =

D(K)E,(K)

Adj M(K)

>

0 and

of Adj M(K),

the

for Z(K) = (Adj M(K))en < 0, so that M(K)

4 C,

ME C.

if E,(O) # 0, Theorem

3.1 applies.

As just above,

for

K =

0,

the E%(O)D(O) all negative preclude M E C; whereas, if M 4 C, by Theorem 3.1, D(0) < 0, and we are assuming that E,(O) = (Adj M),,, using Z(K) as above,

Q(z(0)) = D(O)E,(O)

< 0.

But,

> 0. Then,

as in the proof

of

Theorem 3.1, this implies that either z(0) > 0 or z(0) < 0. Since z(O), > 0, we have z(0) > 0, from which EJO)D(O) < 0, completing

* In

[15]

Motzkin

used the term “copositivt”

to mean what we have called

strictly copositive. Linear

Algebra

and Its Applications

the proof.

3(1970), 295-310

ON

CLASSES

OF

COPOSITIVE

309

MATRICES

We define the class C as follows (the “Motzkin”

Proof of Theorem 4.2.

class) : IM E -C if and only if, for each principal submatrix M* of M, the last column of Adj il1* is positive, then Det M* is positive. It is clear that C has the property

of inheritance.

The theorem

may be restated

as:

C = C”. First,

suppose ME C*.

has a positive

last column,

M* E C*, it suffices

If M* is a principal

submatrix

we wish to conclude

that Det M* > 0.

to show this for M itself.

M is C* of order n -

whose adjoint Since

Now M E C* implies that

1. As in the proof of Theorem

3.2, for z = (Adj M)ell

we have

Q(z)= Pet WWj so that Hence

z > 0, and then C* is contained

ML,,,

1w E C* implies

Q(z) > 0, forcing

Det M > 0.

in C.

Second, suppose that M E C. Then M is C* of order 1; that is, M,,i > 0. We make conclude

the inductive

assumption

that

M is C* of order

that then M is C* of order Y + 1; But,

order Y + 1 are in C*. showing that, standing,

by inheritance,

if M is C* of order n -

Theorem

3.2 applies.

hold, which would contradict ME C*, completing We mention

to us, and which

and

n,

of

this is clearly

to

1, then M E C*.

equivalent

With this under-

If it were that M 4 C*, (i) and (ii) would M E C. Hence, by induction,

a result

parallels (E.

only if each @incifial nonnegative

<

M E C implies

the proof.

finally

THEOREM 4.2

7

that is, that submatrices

of E. Keller,

the result

Keller).

which was communicated

of Motzkin.

A symmetric

matrix

is copositive

submatrix for which the cofactors of the last

has a nonnegative

if and YOW

aye

determinant.

REFERENCES 1 V.

J.

D.

Baston,

Extreme

copositive

quadratic

forms,

Arith.

18(1969),

19(1966),

197-204.

Acta

319-327.

2

L.

D.

Baumert,

Extreme

3 L. D. Baumert, 4 L.

D.

5 R. W. over

Extreme

Baumert, Cottle, convex

Princeton,

to

appear

and in

Pacific J. Math.

forms, forms,

Pacific

II,

communication,

G. J. Habetler, cones,

Dec.

13.

C. E. Lemke, Proc.

Int.

J. Math.

20(1967),

l-20.

1967. Quadratic

Symp.

forms

on Mathem.

semi-definite Programming,

1967.

6 P. H. Diananda, nonnegative,

private

copositive copositive

On nonnegative

Proc.

Cambvidge Linear

forms Philos. Algebra

in real variables SW.

88(1962),

and

Its

some

or all of which

are

17-25.

Applications

3(1970),

295-310

310

R. W. COTTLE, Matrix

7 1’. R. Gantmacher,

G. J. HABETLER,

theory, Vol.

8 A. Garsia, Remarks about copositive ematics,

University

9 M. Hall,

of California,

Jr., Combinatorial

AND

C. E. LEMKE

2, Chelsea, New York,

1959, p. 53.

forms, Working Paper, Department San Diego

theory,

(Dec.

Blaisdell

of Math-

1, 1964).

Publishing

Co., Waltham,

Mass.,

1967. 10 M. Hall, forms, 11 Emilie

Jr., and M. Newman,

Haynsworth

and A. J. Hoffman,

Lineav Algebra 2(1969), 12 C. E. Lemke, Managewwzt 13 T.

Copositive

and completely

Proc. Cambridge Plzilos. Sm. 59(1963),

quadratic

Two Remarks

on Copositive

hfatrices,

387-392.

Bimatrix

equilibrium

points

and

mathematical

programming,

Sci. 11(1965), 681-689.

S. hlotzkin,

14 T. S. hlotzkin,

in

Nat. Bur. Standavds Rep. 1818 (1952), 11-12.

Quadratic

Notices Amer. Math. Sot. 15 T. S. Motzkin,

New York,

positive

329-339.

forms positive 12(1965),

for nonnegative

Signs of minors, in Iwqzcalities,

1968

Linear Algebra and Its Applications

not all zero,

0. Shisha (ed.), Academic

1967.

Received July,

variables

224.

3(1970),

295-310

Press,