ON PRINCIPAL MINORS OF POSITIVE DEFINITE MATRICES

by

S. R. Searle

BU-756-M

September, 1981

Biometrics Unit, Cornell University, Ithaca, New York

Abstract

Simple arguments are given for the inductive proof of the well-known result that all principal leading minors of a (symmetric) positive definite matrix are positive.

It is well known that all principal leading minors of a positive definite (p.d.) matrix (assumed, as is customary, to be symmetric) are positive. Seelye (1958) gives an inductive proof, for the bulk of which we here offer simpler arguments.
~Theor..e..m...: A real symmetric matrix is positive definite if and only if all
its principal leading minors are positive.
Proof: Let A= A' be real and p.d., and for vectors band x and scalars B and A define
(1)

-2-

where x and A have the same number of rows. The inductive proof is based on assum-

ing the positive definiteness of~ and showing that~ is then p.d. if and only if

all principal leading minors of ~ are positive. Because these minors consist of all such minors of~' and 1~1 itself, and because~ is being assumed p.d., we then have only to show that B is p.d. if and only if 1~1 is positive. To do this

we use

(2)

and

(3)

(4)

ll:R2till: that if~ is p.d., then 1~1 > 0.

- -- -- 1B being p.d. means that W1Bw > 0 for all w 0. First take W1 = [0 ~], i.e.,
2·from (1), ~ = Then from (4), ~~~ > 0 ~ ~A2 > 0 ~ ~ > 0 because A is real.

=Second , t ake ~ I

e . '[~~~~~ -1

b-

1

A
-

-~L/

A]
.....

;

.
J..

X1

=

b 1A-l

and

'=
1\

-

~
·-

~~~-l!~J/A......

Then again from (4)

i.e.
Because~ is p.d., 1~1 > 0; and A-lis also p.d., and so b 1 A-~ > 0. Therefore from (5) we have ~ - ~~~-~ > 0 and this together with 1~1 > 0 implies from (2)
..that IBI > 0. Q.E.D•
~<2.<2!..itU: that if 1~1 > 0, then ~ is p.d. 1~1 > 0, along with 1~1 > 0 implies from (2) that~>~·~-~· Then in (3)
-- -- - -- - - - -wI Bw > X I Ax + 2 Ab I X + x.2 (b I A-lb) > (~ + ~-~)'~(~ + ~-~) > 0 because A is p.d.
Therefore B is p.d. Q.E.D.

·~
-3-
As final comment, we note that although the theorem is in terms of principal leading minors it applies to all principal minors. We state this as a corollary.
~: All principal minors of a symmetric p.d. matrix are positive.
+Proof: Let A= A' be p.d .• Then x'Ax > 0 for all x 0 and, for P
~..-...;....,
being a permutation matrix, x'P'APx > 0 also. Therefore P'AP is p.d., and so by the theorem its principal leading minors are positive. But, over the set of all possible P's, the principal leading minors of P'AP are the principal minors of A, which are therefore positive. Q.E.D.
Reference
Seelye, c. J. (1958) Conditions for a positive difinite quadratic form
established by induction. Am. Mathematical Monthly, 65, 355-6.