vol15_pp314-328. 157KB Jun 04 2011 12:05:56 AM

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 15, pp. 314-328, November 2006

ELA

http://math.technion.ac.il/iic/ela

A NOTE ON NEWTON AND NEWTON–LIKE INEQUALITIES FOR
M–MATRICES AND FOR DRAZIN INVERSES OF M–MATRICES∗
MICHAEL NEUMANN† AND JIANHONG XU‡
Abstract. In a recent paper Holtz showed that M–matrices satisfy Newton’s inequalities and so
do the inverses of nonsingular M–matrices. Since nonsingular M–matrices and their inverses display
various types of monotonic behavior, monotonicity properties adapted for Newton’s inequalities are
examined for nonsingular M–matrices and their inverses.
In the second part of the paper the problem of whether Drazin inverses of singular M–matrices
satisfy Newton’s inequalities is considered. In general the answer is no, but it is shown that they do
satisfy a form of Newton–like inequalities.
In the final part of the paper the relationship between the satisfaction of Newton’s inequality by
a matrix and by its principal submatrices of order one less is examined, which leads to a condition
for the failure of Newton’s inequalities for the whole matrix.

Key words. M–matrices, Nonnegative matrices, Generalized inverses, Newton’s inequalities.
AMS subject classifications. 15A09, 15A24, 15A42.

1. Introduction. In a recent paper, Holtz [5] showed that Newton’s inequalities
hold for the class of matrices consisting of the nonsingular M–matrices and their inverses.
Newton’s inequalities involve the (normalized elementary) symmetric functions
that are defined as follows.
Definition 1.1. For x1 , . . . , xn ∈ C and for 1 ≤ j ≤ n, define the symmetric
functions


xi1 · · · xij
(1.1)

Ej (x1 , . . . , xn ) :=

1≤i1 0, for all j = 0, . . . , n.
The following identity can be found in [12] and [11], but it is likely to be found
in earlier literature:
Lemma 2.1. (Rossett [12, Eq. (4)]) For the nonzero scalars x1 , . . . , xn ∈ C, let
x′i = 1/xi , i = 1, . . . , n. Then:

(2.4)
3 Note

En (x1 , . . . , xn )En−j (x′1 , . . . , x′n ) = Ej (x1 , . . . , xn ), 0 ≤ j ≤ n.
that Ej (A) in [6] is equivalent to

n

Ej (A)
j

here.

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 15, pp. 314-328, November 2006

ELA

http://math.technion.ac.il/iic/ela

318

M. Neumann and J. Xu

Recall now that if A ∈ Rn,n is nonsingular, then En (A) = 0. We next develop
the following two lemmas:
Lemma 2.2. Let A ∈ Rn,n be a nonsingular matrix and let {k1 , . . . , kn−1 } be a
set of nonnegative constants. Then
(2.5)

Ej2 (A) ≥ kj Ej−1 (A)Ej+1 (A), 1 ≤ j ≤ n − 1,

if and only if
(2.6)

Ej2 (A−1 ) ≥ kn−j Ej−1 (A−1 )Ej+1 (A−1 ), 1 ≤ j ≤ n − 1.

Proof. We know that if σ(A) = {x1 , . . . , xn }, then σ(A−1 ) = {x′1 , . . . , x′n }. To
show the necessity, we apply (2.4) to each factor in (2.5) and obtain that
2

(A−1 )
En2 (A)En−j

≥ kj En2 (A)En−j+1 (A−1 )En−j−1 (A−1 ),
and so
2
En−j
(A−1 ) ≥ kj En−j+1 (A−1 )En−j−1 (A−1 ).

This yields (2.6) upon replacing n − j with j.
The proof of the sufficiency part can be done by observing the symmetry between
(2.5) and (2.6).
A simple consequence of Lemma 2.2 is that when A is a nonsingular matrix and
satisfies Newton’s inequalities, which corresponds to the case when all the kj ’s
in Lemma 2.2 are equal to 1, then its inverse A−1 also satisfies Newton’s inequalities. This provides a further confirmation to a part of Holtz’s [5] results that Newton’s
inequalities hold for nonsingular M–matrices if and only if they hold for inverses of
nonsingular M–matrices.
We next present a modification of Lemma 2.2 which will allow us subsequently
to examine the fulfillment of Newton’s inequalities by certain generalized inverses of
M–matrices.

Lemma 2.3. Let B ∈ Rr,r be a nonsingular matrix with spectrum σ(B) =
{x1 , . . . , xr } and suppose that there exists a set of constants {k1 , . . . , kr−1 } such that
(2.7)

Ej2 (B) ≥ kj Ej−1 (B)Ej+1 (B), 1 ≤ j ≤ r − 1.

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 15, pp. 314-328, November 2006

ELA

http://math.technion.ac.il/iic/ela

319

Newton–Like Inequalities

For any n > r, set xr+1 = . . . = xn = 0. Then
(2.8)

 2 


r
n
n
j
j−1 j+1
Ej2 (x1 , . . . , xn ) ≥ kj  2 

 Ej−1 (x1 , . . . , xn )Ej+1 (x1 , . . . , xn ),
n
r
r
j
j−1 j+1
for 1 ≤ j ≤ r − 1. In particular, if Ej (B) > 0, for j = 1, . . . , r, then
(2.9)

Ej2 (x1 , . . . , xn ) > kj Ej−1 (x1 , . . . , xn )Ej+1 (x1 , . . . , xn ), 1 ≤ j ≤ r − 1.

Proof. To show (2.8), we first observe that for any 1 ≤ j ≤ r,


xi1 · · · xij
Ej (x1 , . . . , xn ) =

(2.10)

=

=

1≤i1 1/2, note that
(4.5)

c1 =

1 rn − r n − r + 2
·
·

.
2 rn − n n − r + 1

To illustrate the result of Corollary 4.2 let us return to the example of the M–
matrix A given in (1.5) whose Drazin inverse is given in (1.6). Here A has precisely
one zero eigenvalue so that n = 5 and r = 4. Substituting these values in (4.5) yields
that in this example c = 0.8.
We finally remark that if A = sI − B is a singular and irreducible M–matrix,
then A has a {1}–inverse which satisfies Newton’s inequalities. To see this partition
A into


A1,1 A1,2
.
A =
A2,1 A2,2
where A1,1 is (n − 1) by (n − 1). It is known that A has rank n − 1, that A1,1 is a
nonsingular M–matrix, and that
 −1


A1,1 0
−
A =
0
0

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 15, pp. 314-328, November 2006

ELA

http://math.technion.ac.il/iic/ela

Newton–Like Inequalities

325

is a {1}–inverse of A. Next, as Newton’s inequalities hold on A−1
1,1 , by Lemma 2.3 we

conclude, with x1 , . . . , xn−1 , being the nonzero eigenvalues of A− and with xn being
its only zero eigenvalue, that
 2 


r
n
n
j
j−1 j+1
Ej2 (A− ) ≥  2 

 · Ej−1 (A− )Ej+1 (A− )
n
r
r
j
j−1 j+1
≥ Ej−1 (A− )Ej+1 (A− ),
for 1 ≤ j ≤ n − 2. These inequalities continue to hold in the case when j = n − 1 as

En (A− ) = 0.
5. Conditions for the Failure of Newton’s Inequalities. In this section we
explore conditions under which for a matrix A ∈ Rn,n , Newton’s inequalities will fail.
For that purpose let σ(A) = {x1 , . . . , xn } and, for i = 1, . . . , n, denote by Ai the
principal submatrix of A obtained from A by deleting its i–th row and column.
We begin by obtaining a relation between Ej (A) and Ej (Ai ), which can be regarded as the special case of a formula in [5, p. 712]. As earlier in the paper, let
p(x) = det(xI − A) be the characteristic polynomial of A. It is well known, see for
example Horn and Johnson [6, Exercise 4, p. 93], that
n


d
p(x) =
det(xI − Ai ).
dx
i=1

Thus, from the representation of the characteristic polynomial as given in (2.3) when
it is applied to matrices of order n − 1 we obtain that




n−1
n


1
j n−1
1
d
(−1)
Ej (Ai ) xn−j−1 .
(5.1)
n · dx p(x) =
j
n
j=0
i=1
Next, let y1 , . . . , yn−1 be the roots of
(5.2)

d
dx p(x).

Then



n−1


n−1
1 d
·
p(x) =
(−1)j
Ej (y1 , . . . , yn−1 )xn−j−1 .
j
n dx
j=0

Upon equating the coefficients of likewise powers of x in (5.1) and (5.2), we obtain
that for 0 ≤ j ≤ n − 1,
n

Ej (y1 , . . . , yn−1 ) =

1

Ej (Ai ).
n i=1

Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 15, pp. 314-328, November 2006

ELA

http://math.technion.ac.il/iic/ela

326

M. Neumann and J. Xu

Now, according to Niculescu [11, Section 2], Ej (A) = Ej (y1 , . . . , yn−1 ) for 0 ≤
j ≤ n − 1. Hence,
n

Ej (A) =

1

Ej (Ai ), 0 ≤ j ≤ n − 1.
n i=1

We are now ready to prove the following result. For simplicity of notation, we
write Ek,i := Ek (Ai ) for i = 1, . . . , n and k = 0, . . . , n − 1.
Proposition 5.1. Let A ∈ Rn,n . Suppose that for some k = 1, . . . , n − 2, and
for all i = 1, . . . , n,
(5.3)

Nk (Ai ) ≤ 0, Ek−1,i ≥ 0 and Ek+1,i ≥ 0.

Then
(5.4)

Nk (A) ≤ 0,

that is, the k–th Newton inequality fails to hold.
Proof. Again, for j = 0, . . . , n, set Ej := Ej (A). Now from the definition of
Nk (A) in (1.4) we can write that:


n2 Nk (A) = n2 Ek2 − Ek−1 Ek+1
=

=



n



Ek,i

i=1

2

−

 n



Ek−1,i

i=1



n


i=1

Ek+1,i



n


 2

Ek,i − Ek−1,i Ek+1,i
i=1

+




(2Ek,i Ek,m − Ek−1,i Ek+1,m − Ek+1,i Ek−1,m )

1≤i