AbstractWe investigate classes of real square matrices possessing some weakened from of strict diagonal dominance of a real matrix whose diagonal entries are all positive. The intersection of each one of these classes with the set of all real matrices, with nonpositive off-diagonal elements, coincides with the set of all nonsingular M- matrices

Neumann, M.

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A Note on Generalizations of 
Strict Diagonal Dominance for Real Matrices 
M. Neumann 
Department of Mathematics 
The University of Nottingham 
Nottingham NC7 2RD, England 
Submitted by R. S. Varga 
ABSTRACT 
We investigate classes of real square matrices possessing some weakened form of 
strict diagonal dominance of a real matrix whose diagonal entries are alI positive. The 
intersection of each one of these classes with the set of all real matrices, with 
nonpositive off-diagonal elements, coincides with the set of all nonsingular M- 
matrices. 
1. INTRODUCTION 
In [8] Varga showed that many of the generalizations (to the concept) of 
strict diagonal dominance of an n x n complex matrix can, indeed, be based 
upon the theory of nonsingular M-matrices. More recently Plemmons [5] has 
surveyed the numerous characterizations, which can be found in the works 
of authors from different disciplines, for an n X n real matrix, with nonposi- 
tive off-diagonal entries to be a nonsingular M-matrix. 
In Plemmons’s survey these diverse characterizations are grouped into 
classes, such that each class contains several conditions which are equivalent 
for arbitrary real (and square) matrices. Moreover, a directed graph is 
presented there to show the relationships between the classes. (Here we shall 
refer to the classes of characterizations as classes of matrices.) 
Unfortunately, the classification in [5] is incomplete, particularly with 
respect to those classes of matrices which possess some weakened form of 
strict diagonal dominance of a real matrix whose diagonal entries are all 
positive. In fact, the author of [5] makes no secret of this incompleteness, 
LINEAR ALGEBRA AND ITS APPLICATIONS 26:3-14 (1979) 3 
0 Elsevier North Holland, Inc., 1979 00243795/79/04ooo3 + 12$01.75 
4 M. NEUMANN 
pointing out several questions, the answers to which were unknown to him at 
the time of preparing the survey. 
This note grew out of an attempt to answer some of the questions raised 
in [5]. However, as happens on some occasions, we in turn raise certain 
questions, the answers to which are unknown to us at this time. Our main 
results are given in Lemmas 3 and 4 and Propositions 1 and 2. We make use 
of these results to present a refinement of the directed graph in [5] 
mentioned earlier. 
2. NOTATIONS AND PRELIMINARIES 
We shall denote by R” (C”) and by R”,” ((7”) the n-dimensional real 
(complex) space and the set of all n x n real (complex) matrices, respectively. 
The ith entry of a vector x E R” will be denoted by xi or (x)~, and sometimes 
it will be convenient to denote the i,jth entry of a matrix S E R”*” by (S&. 
For any two nonnegative integers m and 1 with I> m, [m, I] is the set of 
integers m, m + 1,. . . , 1. 
For S ER”,“, the symbols Sr,a( S) and p(S) denote, respectively, the 
transpose, the spectrum and the spectral radius of S. If x ER” and each 
component of x is nonnegative (positive), we shall write x > 0 (x > 0), and if 
each element of a matrix S E R n,n is nonnegative we shall write S > 0. A 
diagonal matrix will be called positive if all its diagonal entries are positive 
numbers. If S E C”,“, then IS) will denote the real matrix whose entries are 
the moduli of the entries of S. 
Next consider the subset of R”,” defined by 
Z”*” = { S=(sii)~R”%sii<O, i#j}. 
Any matrix S E Z”,” can be expressed in the form 
s = tl- u, (2.1) 
where U > 0 and t > 0. A matrix S E Z”,” is called an M-matrix if it has a 
representation (2.1) with t > p( U). If S is a nonsingular M-matrix, then for 
any representation (2.1) of S we have that t > p( U) (e.g. [5]). It is well known 
that S E Z”,” is a nonsingular M-matrix if and only if S is nonsingular and 
S - ’ > 0 (see [2]). Th is characterization is often used in the literature as the 
definition of a nonsingular M-matrix (see, for example, [9, Chapter 31). We 
shall denote by P the class of all nonsingular M-matrices. For S = (sii) E R”*“, 
GENERALIZATIONS OF STRICT DIAGONAL DOMINANCE 5 
the comparison matrix %Q( S) for S is the real matrix given by 
(m( S))ij = 
i 
lsiil if i=i, 
-jsiil if i#j. 
Next let S = ( sii) E C”~“. Then S is strictly diagonally dominant if 
l’iil > igI I’iily i E [In]. 
j#i 
In [l], Beauwens defines S to be lower semistrictly diagonally dominant if 
Beauwens defines S to be semistrictly diagonally dominant if for some 
permutation Q, the matrix QSQ’ is lower semistrictly diagonally dominant. 
It follows that every strictly diagonally dominant matrix is semistrictly 
diagonally dominant. That the converse of this statement is not true is 
illustrated by taking 
SC 1 1 ( 1 1 2’ 
We shall adopt here the same letters of the alphabet used in [5] to denote 
classes of matrices possessing certain given properties. For convenience we 
display here those classes of matrices which are most relevant to this note. 
L: The class of matrices S = (sii) E R n,n with the following properties: 
for each S EL there exists a vector x > 0 such that 
sx 2 0 
and such that if for some index i, E [ 1, n], 
(2.2) 
(SX)i” = 0 (2.3) 
6 
there exist indices 
M. NEUMANN 
such that 
P-5) 
and such that 
(SX)& > 0. (2.6) 
M: The class of matrices S = (sii) E R n,n with the following properties: 
for each S EM there exists a vector x > 0 such that Sx > 0 and such that 
I 
x siixi > 0, i E[l,n]. 
i=l 
P-7) 
M': The class of matrices S E R”,” with the property that for each 
matrix S E M’ there exists a permutation matrix Q such that QSQT E M. 
N: The class of matrices S E R T” with positive diagonal entries such 
that for each matrix S EN there exists a positive diagonal matrix D such that 
the matrix SD is strictly diagonally dominant. 
If C, and C, are any two classes of matrices in R”,“, by C,+C, we shall 
mean that a necessary condition for a matrix to be in C, is that it be in C,. 
3. THE MAIN RESULTS 
In [4] Moylan showed that a real matrix S with all its diagonal entries 
positive is in N if and only if m(S) E P. Our first result shows that in addition 
to the list of characterizations in [5], N can be further characterized by 
properties which relate to semistrict diagonal dominance. 
LEMMA 1. Let S be a real matrix with all its diagonal entries positive. 
Then the following statements are equivalent: 
(i) For some positive diagonal matrix D, the matrix SD is semistrictly 
diagonally dominant. 
(ii) S EN. 
GENERALIZATIONS OF STRICT DIAGONAL DOMINANCE 7 
Proof. Clearly (ii) implies (i), so we show that (i) implies (ii). Let Q be a 
permutation matrix such that the matrix QSDQ’ is lower semistrictly diago- 
nally dominant. By the Corollary in [l, p. 1111 it follows that QEm( SD)Q T= 
‘%!( QSDQ ‘) E P, and hence by the properties of P, !?J2( SD) E P. Finally, since 
D - ’ is a positive diagonal matrix, we have that 
92(S) = %!(SD)D-’ E P 
by [3, Theorem 4.91, and hence S E N by [4]. H 
Lemma 1 has several simple corollaries. We only give here two of the 
possible ones. 
COROLLARY 1. Let S E C”,“. Then the following statements are equiv- 
alent: 
(i) There exists a nonsingular diagonal matrix D E C”~” such that the 
matrix SD is semistrictly diagonally dominant. 
(ii) For some nonsingular diagml matrix D EC”,“, the matrix SD is 
strictly diagonally dominant. 
Proof. That (ii) implies (i) is obvious. (i) implies (ii): Since SD is 
semistrictly diagonally dominant, it follows at once that the diagonal entries 
of the matrix S are all nonzero. Thus, since 1 SD I= ) S 1) DI, the matrix ) S I 
satisfies the conditions of statement (i) in Lemma 1. But then there exists a 
positive diagonal matrix, say D’, such that the matrix IS ID’ is strictly 
diagonally dominant and hence SD’ is strictly diagonally dominant. n 
Corollary 1 leads us to the conclusion that the concepts of strict diagonal 
dominance, lower semistrict diagonal dominance and semistrict diagonal 
dominance for a complex matrix are equivalent to each other up to a 
multiple by a nonsingular diagonal matrix D E (7’3”. We wish to remark that 
Lemma 1 and Corollary 1 could also be derived from Beauwens’s results in 
[l] or from Varga’s results in [8]. 
The second corollary is more of an outcome of Corollary 1. 
COROLLARY 2. Suppose that S E C”,” is semistrictly diagonally aTomi- 
nant. Then the point Jacobi and the point Gauss-Seidel iteration matrices 
associated with S are both convergent. 
Proof. Using similarity arguments, the proof easily follows from [9, 
Theorem 3.41 and Corollary 1. 1 
8 M. NEUMANN 
One of the principal goals of this note is to show that M*L. This will be 
accomplished in Lemma 3 below. Beforehand, we offer an observation which 
may be helpful in determining whether a matrix S E R W” is in L and which 
generalizes a result of Vandergraft in [6]. 
LEMMA 2. Let S=(sii)~R”,“, and suppose that there exists a rwnzero 
vector x E R n such that 
CA= {kE[l,n]:(Sx)k#O} #0. 
Then for each index iE[l,n]\O there exist indices &...,&E[l,n], with 
i,ECJ, such that the product 
sii,si,i,’ . * S&_,i, # 0 
if and only if the vector Sx does not belong to any proper subspace of R”, 
spanned by unit coordinate vectors, which is invariant under S. 
The proof of Lemma 2 is not dissimilar to the proof of [6, Theorem 4.11 
and is therefore omitted. 
LEMMA 3. Suppose that S = (sji) E R “,” is in M. Then S EL. More 
specifically, if x > 0 is a vector such that Sx > 0 and such that (2.7) holds, 
then for each index i, E [ 1, n - l] f or which (2.3) holds, there exists a strictly 
increasing sequence of integers (2.4) such that (2.5) and (2.6) are satisfied. 
Proof. If SX > 0, then S EL and there is nothing to prove. Suppose then 
that (2.3) holds for some index i, E [l,n - 13. If sion #O, then since (Sx), >0 
by (2.7), Equations (2.5) and (2.6) hold with r= 1, and so S EL. Assume 
therefore that sifl = 0. By (2.7) 
5 si,pj > 0, 
j=l 
and so by (2.3) there exists an index i, E [iO + 1, n - l] such that 
Si”i, # 0. (3.1) 
If (SX)~, >O, then again (2.5) and (2.6) hold with r= 1, and hence S EL. 
Suppose then that 
GENERALIZATIONS OF STRICT DIAGONAL DOMINANCE 9 
and apply to the index i, similar considerations to those applied to the index 
i,, and so on. Since the sequence of indices { ik} thus generated is strictly 
increasing, the process must terminate after a finite number of steps, say r, 
with i, = n if at no previous stage sik_ ,” # 0 or s&_,~ = 0 but ( SX)~, > 0, 
k E [ 1, T - 11. Thus S E L and the proof is complete. n 
The next result will allow us to determine the relationship between the 
classes N, M, M’ and L. 
LEMMA 4. Let S = (sii) E R”,“. Then S EL if and only if fm every 
permutation matrix Q, the matrix 
QSQ’ E L. 
Proof. The proof of the “if’ part is obvious. 
“Only ir’: Let Q be an nth order permutation matrix, and let p be the 
permutation function associated with Q. Let x > 0 be a vector such that (2.2) 
through (2.6) hold (with respect to x). Set X~EQX and S,=QSQ’. Then 
x0 > 0, and since SQxq = QSx, we have from (2.2) that SoxQ > 0 and that the 
set 
{ jE[l,n]:(SgXO)i>O} #0. 
Suppose that for some index ja E [ 1, n], 
(s&& = 0. 
Then since 
(Sxho) = (SOT&y io) = (Sp&,~ 
it follows from (2.3) through (2.6) that there exists indices i,, . . . ,i,~[l,n] 
such that 
and such that 
(SX)i, > 0. 
10 M. NEUMANN 
But then the elements 
of the matrix SP are all nonzero, and 
This concludes the proof. 
We thus have the following result: 
n 
PROPOSITION 1. In R”*“, N=+M*M’=+L. 
Proof. That N=+M*M’ follows immediately from the definitions of 
these classes, and that M’+L follows from Lemmas 3 and 4. a 
REMARK. Note that if S EM’, then QSQ’E M’ for any permutation 
matrix Q. Hence both classes M’ and L are closed under same reordering of 
rows and columns of members in these classes. The class M does not possess 
a similar property, as illustrated by taking 
Next, we investigate the relationship of the classes M and M’ to some of 
the other properties of matrices studied in [5]. 
PROPOSITION 2. Let S E R”,“. 
(a) For 
properties : 
(i,) S + CWZ 
(i$) There 
matrices E, 
S to belong to M, S need not have any of the following 
is nonsingular for all a > 0. 
exist a permutation matrix Q and lower and upper triangular 
and E, respectively, with all their diagonal entries positive, such 
that QSQ T = E,E,. 
(iii,) Every splitting of S into S = R - T, where T > 0 and R i.s rwnsingular 
with R -’ > 0 (i.e., the splitting S = R - T is regular), bus p(R -‘T) < 1. 
(b) For S E R”,” to have any me of the following properties: 
(i,,) S is nonsingular and S - ’ > 0, 
GENERALIZATIONS OF STRICT DIAGONAL DOMINANCE 11 
(iib) there exists a positive diagonal matrix D such that the matrix SD+ DST 
is positive definite, 
S need not be in M. 
(c) For S to h&ve any one of the following properties: 
(4) (see (4JY 
(ii,) each eigenvalue of S has a positive real part, 
(iii,) for each k E [l, n], the sum of all the principal minors of S of order k is 
positive, 
(iv,) S = E,E,, where E, and E, are 1owe-r and upper triangular matrices 
respectively, with all their diagonal entries positive, 
S need not be in M’. 
Proof. (a): Consider the matrix 
s= -d 4 -d, 
I 
-2 
0 -2 3 I 
and let x = (1, 1, l)T. Then Sx > 0. Moreover the matrix S and the vector x 
satisfy (2.7), showing that S EM. Since S is singular, S cannot satisfy (i,) or 
(iiJ Consider the splitting of S into 
Since 
R-l=1 
224 
I 32 16 0 71 164 28 14 01 > 
i 0 71 0 0 61 =R-T. 
the splitting S = R - T is regular. Now a(R -IT) = { 1, a, 0}, and hence S does 
not satisfy (iii,). 
(b): Let 
s=(-: A). (3.2) 
12 M. NEUMANN 
Then S satisfies (i,J but S 4 M. Next suppose that 
1 -; -1 
3 
s= -g 1 1. 
I 
(3.3) 
<--1 -1 1 
It is easy to verify the matrix S + ST is positive definite, so that S satisfies 
(iit,). Assume now that there exists a vector X= (x,,x,,~,)r >0 such that 
sx > 0. (3.4) 
and such that (2.7) is satisfied for i~[1,3]. Then from (3.3), (3.4) and (2.7), 
the component of x have to satisfy the following inequalities: 
and 
x3 > x1 + x2. (3.7) 
Inserting (3.7) in (3.5) h s ows that the components of x have also to satisfy 
But then inserting (3.6) in (3.8) we see that 
x >% 1 36 1’ 
which is impossible. Thus S G? M. 
(c): Let S be as in (3.2). Then S satisfies (it,), but S @M’. Next let 
s=( -: 2). 
Then S satisfies the properties in (ii,) through (iv,), but clearly S @M’. This 
completes the proof. n 
GENERALIZATIONS OF STRICT DIAGONAL DOMINANCE 13 
FIG. 1. 
The results of Propositions 1 and 2 and of Theorem 1 in [5] yield the 
directed graph in Fig. 1 for the relationship between the classes of matrices 
introduced in [5]. 
We remark that the implication K=+G is due to Vandergraft [I. 
COROLLARY 3. In fi n’n, 
p = N ” Z”,” = M ” Z”,” = M’ ” Z”,” = L n Z”,“. 
Proof. The proof follows from Proposition 1 and from Theorem 1 in [5]. 
n 
4. OPEN QUESTIONS 
(1) If all the principal minors of S E R “sn are positive (i.e., S E A), does 
S E M' or not? 
(2) If S satisfies property (iii,) (i.e., S EI), does S EM’ or not? 
We remark that C. R. Johnson and the author know that the answer to 
both questions is in the affirmative for n < 3. Indeed, permuting the first row 
and first column with the second row and column, respectively, in the matrix 
14 M. NEUMANN 
S given by (3.3) yields a matrix in M [with respect, for example, to the vector 
x = (1,2,3 + E)~ for E > 0 sufficiently small]. 
The author wishes to thank Professor R. I. Plemmons of the University of 
Tennessee at Knoxville for his encouragement and support in writing this 
note. Professor R. Beauwens and D. A. Burgess are thanked for their help 
and interest. Finally, the author wishes to thank Professor R. S. Varga for his 
comments concerning the original draft of this note. 
REFERENCES 
It. Beauwens, Semistrict diagonal dominance, SIAM J. Numer. Anal. 13:109-112 
(1976). 
K. Fan, Topological proofs for certain theorems on matrices with nonnegative 
elements, Munutsh. Math. 62:219-237 (1958). 
M. Fiedler and V. Ptak, On matrices with nonpositive off-diagonal elements and 
positive principal minors, Czech. Math. J. 12:382-400 (1962). 
P. J. Moylan, Matrices with positive principal minors, Linear Algebra Appl. 
175358 (1977). 
R. J. Plemmons, M-matrix characterization I: Nonsingular M-matrices, Linear 
Algebra Appl. 18: 175-188 (1977). 
J. S. Vandergraft, A note on irreducibility for linear operators on a partially 
ordered finite dimensional vector space, Linear Algebra Appl. 13:139-146 (1976). 
J. S. Vandergraft, Applications of partial orderings to the study of positive 
definiteness, monotonicity, and convergence of iterative methods for linear sys- 
tems, SIAM J. Numer. Anal. 9:97-1&l (1972). 
R. S. Varga, On recurring theorems on diagonal dominance, Linear Algebra Appl. 
13:1-9 (1976). 
R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N.J., 
1962. 
Received 29 October 1977; revised 15 May 1978 


A note on generalizations of strict diagonal dominance for real matrices 

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A note on generalizations of strict diagonal dominance for real matrices

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