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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.
i
n
ir
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.
i
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Uncl«8«lfled
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mm a*H »• mmmä ■*■» m» »mmH —ßni it tl—»tl»»0 I. OHMMATIN« ACTIVITY (CMpMato
Stanford University Operations Research House, Stanford, Calif.
t«. MMONT HCUNITV C LMSIFICATION
unclassified •• «neu»
t. MPORT TITLI
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.
N-00014-67-A-0112-0011 a pnoJier NO.
NR-047-064
7*. NO. OP Nirs
6 t*H)
Technical Report 67-1
**■ STHS*.*.f^0,,T N0W Mar «Aw
Technical Report No. 67-1
whaw fM< aar *• aaaljnW
I» «VAILABIUTV/UMITATION MOTICKt
Distribution of this document is unlimited
n. tup )TU
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II. •PONtONINa MILITARY ACTIVITY LOglStlCS and
Mathematical Statistics Branch, Mathematical Sciences Division, Office of Naval Research, Washington, D. C. 20360
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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