I I I I

ON POSITIVE PRINCIPAL MINORS

BY

George B. Dantzig

TECHNICAL REPORT NO. 67-1

January 1967

Operations Research House Stanford University Stanford, California

Research of G. B. Dantzig partially supported by Office of Naval Research, Contract ONR-N-OOO14-67-A-0112-0011, U. S. Atomic Energy Commission, Contract No. AT(04-3)-326 PA //IS, and National Science Foundation Grant GP 6431; reproduction In whole or In part for any purpose of the United States Government Is permitted.

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ON POSITIVE PRINCIPAL MINORS

by

George B. Dantzlg 1)

r When a matrix is symmetric, the property of having all Its

principal minors positive, Is equivalent to being positive definite.

When a matrix Is non-symmetric this Is no longer true. For example

(1) -{:■: has all positive principal minors, but

2 2 xMx " ^ " 7xix2 + «2 * 0 for x " (xi»x2^ " (1»1)

The converse, however, as shown In Gale-Nikaido [i ], Cottle [2 ]

Is true:

Theorem 1; The class with positive principal minors properly Includes

those which are positive definite.

Proof; Assume M Is positive definite, M Is non singular for If not,

then there would exist x - x t 0 such that Mx » 0; yielding

x Mx - 0, a contradiction. It follows that every principal submatrlx

of M Is also nonslngular.

1) The author acknowledges leads suggested by David Gale.

I :;

Partltlop M so that

M M1C

r a mm.

choose x - [Y ,-1] where Y M, - r. Then

*- * , * x Mx - [Y ,-1] M1 c

mm.

r * Y

-1

(a - Y c)> 0 mm

But det Mlc

- det v *

r a m TBBim

0 a • ■> IBB

- Y c (det MJU - Y c)

x mm

Thus det M and its largest order principal minor have the same sign.

Inductively, since first-order principal minors are positive, so must

the' second order ones, etc., up to the highest order. Finally,

example (1) shows that the Inclusion Is proper.

Although positive definite matrices M do not comprise the entire

class of positive principal minors, they can be used to generate a

larger class by multiplying M by diagonal matrices on the right

and left' to form DME. For example.

i

■:

7 0

.0 1.

1 -1

.0 1,

1/7 0

. 0 ij

positive definite

1 -7

Lo IJ positive principal minors but not positive definite

r

r

n

r r

It Is not difficult to show that for 2x2 matrices the entire

positive principle minor class can be so obtained from the positive

definite class. The following is easily seen.

Lenma; Let D - [d ], E - te ] be diagonal matrices with the

property that d-.e > 0, then for any M the sign (+, -, or 0)

of any principal minor of DME is the sarae as that of M. If M

is positive definite, then DME has positive principal minors.

Theorem 2; If M has positive principal minors and there exist

diagonal D and E such that M - D~SlE" is positive definite,

then there exists a diagonal matrix F such that F MF is positive

definite.

— — T Proof: If M is positive definite so is AMA for any nonsingular — T T — T A (since x(AMA )x - yMy > 0 where x j* 0 and y - xA j4 0). We

consider AMAT - AD"1ME~1AT and choose A so that AD-1 - (E^A1)"1

T or A A - ED. Thus A - fa.i.J could be chosen as a diagonal

matrix such that a^ -+-/d e . Then F - [fl1] is the

,-l.T diagonal matrix F - E A where f - + /d-./e .

Theorem 3; The characteristic roots of (any) M are the same as

F~htF for (any) nonsingular F. '

2) A well known result of matrix theory.

I

Proof: By definition, A is a characteristic root of M if there

exists an x j* 0 such that Mx ■ Xx. Setting Fy ■ x then y »* 0

if x j 0 and we have MFy - AFy or (F~ MF) y - Xy so that X

is a characteristic root of (F #F) also.

Theorem 4; If M is positive definite, the real part of every

3) characteristic root Is positive.

Proof; Let Mx - Xx, x i* 0 and let x - u + iv, X - R + iS, then

by substitution and equating the real and imaginary parts

Mu - Ru - Sv (u,v) + 0

Mv - Rv + Su

Now R > 0 follows from

•IU. i^ 2 2 0 < u Mu + v Mv - R (u + v )

The following is known, see Gale - Nikaido [ 1 ].

Theorem 5; If M has positive principal minors, then every real

characteristic root is positive.

3) A well known result in matrix theory that is reviewed here for

non-symmetric M.

!

!

r

r r

r

Proof; The characteristic equation Is obtained by setting

det [M - XI]

this yields

(-X)m + C.C-X)1""1 + C- (-A)"1"2 + ... + C - 0 i. i m

I

where C Is the sum of the J-th order principal minors. For

matrices with positive principal minors, C.> 0. It Is not possible

that A _<_ 0 because then all terms above would be non-negative

and the last term positive so that the left hand side would be

strictly positive, a contradiction.

Theorem 6 CKalman*): There exist matrices M with positive principal

minors that have characteristic roots not all of which have

positive real parts; for such matrices no transformation M - DME

exists with diagonal matrices D and E such that M is positive

definite.

In a letter to D. Gale dated 9 July 1962, Rudolf E. Kaiman showed that

10/3 - 4 - /ll.l

8/3 1/3 - / 1.1

_-/899.1-/89.1 30 -

had positive minors but had complex characteristic roots with negative real parts.

5

I

Proof: If such D and E did exist» then by Theorem 2, an F would

exist such that FMF Is positive definite. By Theorem 4 FMF~

would have to have characteristic roots with positive real parts.

By Theorem 3 the same would be true for M. However M in the

example below does not have this property.

M

1-10

1 1 17

4 0 1 •

The first order principal minors are 1, 1, 1 whose sum C - 3,

its second order ones are 2, 1, 1 whose sum is C» ■ 4. Its third

order one is C_ ■ 70. Thus M has positive principal minors. Its

characteristic equation is

or

X3 - C X2 + C A - C3 - 0

A3 - 3X2 + 4A - 70 - 0

which factors into

(X - 5)(X + 2X + 14) - 0 j ■

The characteristic roots are

ll " +5, x2 " ":l + 1^3, A3 " "1" i*^3 I

Since the real parts of X. and X- are negative, this establishes that

the class of positive principal minor matrices form a larger class

♦

r . *•

r n r

r

than those generated from the positive definite: ones by simple

rescaling of the rows and columns.

In mathematical programming one seeks p-vectors x,y „> 0

that satisfy y - Mx + q such that the products x.y. - 0 for

1 • 1,2,.. »p. The latter conditions may be replaced by the

T T T minimum value of xy-xMx+xq, a quadratic function which is

convex if and only if M Is positive definite. Certain solution

procedures based on convexity [ 2 ], [ 3 ] > [ 4 ] turn out to be also

valid even when M has positive principal minors. This leads to

the speculation that by a simple change of units of x,y one could

obtain a new system y • Mx + q, (yjX^O, yJc.- 0 for i - l,..,p

-T-- -x in which the quadratic function x Mx + x q is convex. However

we have shown that this is not always possible to do. The class

of positive principal minor matrices does not appear to be a

trivial extension of positive definite matrices. Solution

techniques also valid for the latter have somehow gotten around

the difficulties of local optimallty usually associated with

nonconvex programming problems.

An open question posed by Gale and Kaiman and closely related

to considerations found in a paper by Arrow and McManus [5] is the

following:

Suppose for all diagonal matrices D such that d.. >0, that

M has the property that the real part of every characteristic

root of MD is positive, does this imply there exists a D-D0

such that D0M is positive definite?

I

I

REFERENCES

[1] Gale, D. and Nakaido, H., "The Jacoblan Matrix and Global

Univalence of Mappings," Math. Ann. 159 (1965), 81-93.

[2] Cottle, R., "Nonlinear Programs With Positively Bounded

Jacoblans," Ph.D. Thesis, University of California, Berkeley,

March 1964.

[3] Lemke, C. E., "Bimatrix Equilibrium Points and Mathematical

Programming," Management Science 11 (1965), 681-689.

[4] Dantzig, G. B. and Cottle, R. W., "Positive Semi-Definite

Matrices and Mathematical Programming," University of

California, Berkeley, Operations Research Center Report 63-18,

13 May 1963. Also in Non Linear Programming. (J. Abadie, ed.)

North Holland, Amsterdam, 1966.

[5] Arrow, K, J. and McManus, M., "A Note on Dynamic Stability,"

Econometrica 26 (1958) 448-454.

[6] Kaiman, R., Letter to D. Gale dated 9 July 1962 on stable

matrices.

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On Positive Principal Minors

«. eneiuPTivt NOTU fiyp« 3 nSSt mt SSESiw 3B5J

Tsrhnlral Bsnnrr s. MWtmfäTCSrSSCmSiTSSm, tmMtO "~~

Dantzig, George, B.

«. MPORTOATt January 1967

• •. CONTNACT ON «KANT NO.

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Technical Report 67-1

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11. ABSTRACT The relation between matrices with all principal minors positive and

positive definite matrices is explored in the non-symmetric case. It Is shown that simple rescaling of rows and columns is insufficient to transform the former into the latter. As a consequence it appears that the class of problems that can be solved by complementary pivot theory of mathematical programming has been non-trlvlally extended beyond the convex case represented by linear and quadratic programming and takes its place along with another Important extension of Lemke and Howson for the matrix associated with bl-matrlx games.

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