AbstractIntegral matrices A for which the system Ax=b will have an integral basic solution are considered. Results for these matrices are presented which parallel results concerning other matrices for which the system has integral solutions

Bevis, Jean H.

Hall, Frank J.

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Some Classes of Integral Matrices 
Jean H. Bevis and Frank J. Hall 
Department of Mathematics 
Georgia State University 
Atlanta, Georgia 30303 
Submitted by George P. Barker 
ABSTRACT 
Integral matrices A for which the system Ax = b will have an integral basic 
solution are considered. Results for these matrices are presented which parallel results 
concerning other matrices for which the system has integral solutions. 
1. INTRODUCTION 
Consider systems of equations Ax = b where A and b are integral. Those 
matrices A for which integral solutions exist, as well as those for which all 
basic solutions are integral, have been characterized [2, 5, 9, 121. The case for 
basic solutions is important in linear programming, and is related to the 
unimodularity of A. An excellent summary of these results is given by Stephen 
Bamett in [l, Chapter 71. In this paper we introduce three new classes of 
matrices for which consistent systems Ax = b will have at least one integral 
basic solution. We establish relationships between these classes and four 
classes of integral matrices which have been previously studied. The classes 
considered include unimodular matrices, matrices whose columns form a 
unimodular set, and matrices which have an integral (lkinverse. Summaries of 
previous results are given along with new results in order to exhibit parallels 
between the various classes. 
2. DEFINITIONS 
Let A denote an m X n rank-r integral matrix, and let b denote an integral 
column vector. Unless stated otherwise this notation will be used throughout 
the paper. It is known [2, Theorem 41 that Ax = b has an integral solution for 
all integral b in the column space of A if and only if A has an integral 
LINEAR ALGEBRA AND ITS APPLZCATZONS 48:473-483 (1982) 473 
0 Elsevier Science Publishing Co., Inc., 1982 
52 Vanderbilt Ave., New York, NY 10017 00243795/82/070473-11$2.75 
474 JEAN H. BEVIS AND PRANK J. HALL 
{l>inverse, that is, an integral matrix X such that AXA = A. It is also known 
that A wiIl have an integral {l>inverse if and only if the gcd of all r x r 
subdeterminants of A is 1 [3, Theorem 21, or if and only if the Smith canonical 
form of A contains the r X r identity submatrix Z, [2, Theorem 31. This last 
characterization was first established for the case where A has full row rank 
by Smith [9]. 
Even when the system Ax = b has an integral solution, most solutions may 
not be integral, since the null space of A may contain nonintegral vectors. 
There are circumstances where all basic solutions are integral. A basis for A is 
an m X r submatrix of A consisting of T linearly independent columns. If 
Ax = b, then x is a basic solution if there is a basis B for A such that the 
nonzero entries of x correspond to columns of B. 
The term unimodular was first applied to nonsingular integral matrices 
whose determinants are &- 1. The term was then extended to m X n rank m 
matrices all of whose m X m subdeterminants are 0 or f 1. In general an 
m X n rank-r integral matrix is said to be unimodular if for every basis B of A, 
the gcd of all r X r subdeterminants of B is 1. Let Z(A) denote the set of all 
integral b in the column space of A. Then in any of the above cases, A is 
unimodular if and only if all basic solutions of Ax = b are integral for all 
b E Z(A) [lo, 121. 
This concept is generalized by the Dantzig property. An m X n rank r 
integral matrix A is said to have the Dantzig property (or the columns of A 
form a unimodular set) if in each submatrix A, of r rows of A, the nonzero 
r X r subdeterminants of A, have the same absolute value [6, (4.4)]. Let 
Z,(A) denote the set of all integral linear tiombinations of the columns of A. 
Then A has the Dantzig property if and only if all basic solutions of Ax = b 
are integral for all b E Z,(A) [6, (3.9)]. 
Let UNM denote the collection of unimodular matrices, let DAN denote 
the collection of matrices with the Dantzig property, and let INT denote the 
collection of all integral matrices which have an integral (l>inverse. Consider 
the question of when a matrix with the Dantzig property will also be 
unimodular. Now if A E INT, then Z,(A) = Z(A), and one is led to the 
conclusion that INT n DAN = UNM. That is, a Dantzig matrix will also be 
unimodular if and only if it has an integral {l>inverse. A slightly different 
characterization of the Dantzig property is as follows: A E DAN if and only if 
all basic solutions of Ax = b are integral whenever b is such that there is at 
least one integral basic solution [5, (1.5)]. In light of this characterization of 
DAN, the above question leads one to consider matrices which have the 
property that the system Ax = b has an integral basic solution for all b E Z(A). 
We will say that such matrices satisfy the basic solution property and denote 
the collection of all these matrices by BSP. Related to this class, we define the 
class WRSP of matrices which satisfy the weak basic solution property, 
INTEGRAL MATRICES 475 
TABLE 1 
there is an integral there is an integral all basic solutions 
solution basic solution ofAx=bare 
to Ax = b. toAx=b. integral. 
For alI 
b E Z,(A): 
For all 
bEZ(A): 
M(Z) WBSP 
INT BSP 
DAN 
UNM 
namely, the system Ax = b has an integral basic solution for all b E Z,(A). 
The above characterizations can be summarized as in Table 1 where M(Z) 
denotes the class of all integral matrices. To use Table 1 combine the phrase 
at the beginning of a row with that at the top of a column to obtain the 
characterization of the class appearing in that row and column. 
The classes INT, DAN, and UNM have been characterized by subde- 
terminants. For an m X n rank m integral matrix, A E DAN if and only if all 
nonzero m x m subdeterminants have the same absolute value, and A E UNM 
if and only if all nonzero m X m subdeterminants are f 1. Hence for a 
Dantzig matrix A with full row rank, A E UNM if and only if there is one 
such subdeterminant with absolute value 1. In general we define a rank-r 
integral matrix to be subunimodukzr if it contains at least one r X r subde- 
terminant with absolute value 1. The collection of all such matrices is denoted 
by SUB. This generalizes the concept of strong unimodularity introduced by 
Truemper and Chandrasekaran [ll]. An m X n rank-r matrix A is said to be 
strongly unimodular if every basis of A contains an r X r submatrix with 
determinant k 1. We denote the collection of all such matrices by SUM. 
3. CHARACTERIZATIONS 
The containment relations between the various classes are displayed in 
Figure l(a). Figure l(b) shows the relations when one restricts the classes to 
matrices with fnll row rank. 
That SUB c BSP is shown in Lemma 1 below. The other containment 
relations are clear from the definitions and characterizations given in the last 
section. A few examples suffice to show that these collections are distinct. 
Let L = (2 3), M = (1 2), 
1 
1 0 2 2 -1 
476 JEAN H. BEVIS AND FRANK J. HALL 
M(Z) M(Z) 
/ \ / \ 
INT WBSP INT WBSP 
\ / \ \ / \ 
BSP DAN BSP DAN 
/ \ / \ 
SUB UNM SUB I 
\ / \ 
SUM UNM = SUM 
(4 0) 
FIG. 1. 
Then M E SUB and U E UNM, but M @ UNM and U 65 SUB. Also L E INT 
but L GE BSP. Finally it is clear that N P SUB and N P UNM, but we can 
show that N E BSP. Note that Z(N) = 2’ = S, U S, U S,, where Z2 is the 
Z-module of all ordered pairs of integers written as columns, S, is the 
submodule of all such ordered pairs whose first coordinates are even, S, is 
the submodule of all such ordered pairs whose second coordinates are even, 
and S, is the submodule of all such ordered pairs whose coordinates have the 
same parity. Now the first two columns of N generate S,, the third and fourth 
columns of N generate S,, and the fourth and fifth columns of N generate S,. 
Thus N E BSP. The remaining distinctions are either clear or can be obtained 
by combining the above examples with the results of Theorem 2 below. 
LEMMA 1. SUB c BSP, that is if A is subunimodular, then it also has the 
basic solution property. 
Proof. Assuming A has an r X r nonsingular unimodular submatrix, say 
A,, there exist permutation matrices P and Q such that 
PAQ=$ :. 
( i 
Then, since A has rank r, we have 
for some M and N. Since A, is nonsingular unimodular, M and N must be 
integral. Now suppose the system Ax = b, b integral, is consistent. Note that 
INTEGRAL MATRICES 
Ax = b if and only if 
Al 
MA1 
477 
(2) 
Partition conformably 
Q*x=(;;) and Pb= ;: . 
i 1 
Then, Equation (2) yields 
A+, + A,Nx, = b, (3) 
and 
MA,x, + MA,Nx, = b,. (4) 
Equation (4) says that Mb, = b,, so that we need only to find solutions of (3) 
to obtain solutions of (2). We then simply observe that 
satisfies (3), and so 
is an integer basic solution of the original system. Thus, A has the basic 
solution property. n 
The following result states that the collections displayed in Figure 1 form 
a meet semilattice with respect to set inclusion, that is: 
THEOREM 1. (i) BSP = INT n WBSP. 
(ii) UNM = BSP n DAN = INT n DAN. 
(iii) SUM = SUB n UNM = SUB n DAN. 
478 JEAN H. BEVIS AND FRANK J. HALL 
Proof. (i): That BSP c INT r-l WBSP is clear from the inclusion relations. 
Now A E INT implies that Z,(A) = Z(A), and thus A E INT n WBSP implies 
that A E BSP. Part (ii) is obtained similarly, and the proof of part (iii) is given 
after that of Theorem 2. W 
Each of the classes of Figure 1 is characterized in terms of integral 
full-rank factorizations in Theorem 2. That such factorizations exist is given in 
[2, Lemma 21. Note the similarity of these characterizations for classes on the 
same diagonal (from upper right to lower left) of Figure l(a). Related 
characterizations of UNM are given by Truemper in [lo]. 
THEOREM 2. Let A = FG be an integral full-rank factorization of A. 
Then: 
(i) A E WBSP @ G E WBSP. 
(ii) A E DAN e GE DAN. 
(iii) A E INT a F E INT and G E INT. 
(iv) A E BSP a F E INT and G E BSP. 
(v) AEUNM a FEINTandGEUNM. 
(vi) AESUB CJ F,GESUB. 
(vii) A E SUM * F, G E SUM. 
Furthermore, for any matrix F with full column rank, F E INT = F E BSP 
e FeUNMandFESUB = FESUM. 
Proof. For matrices with full column rank the characterizations of INT 
and SUB in terms of subdeterminants are identical to those of UNM and 
SUM respectively. Hence the “Furthermore” portion holds. 
For part (i) suppose that A E WBSP and that b E Z,(G), thus there is an 
integral vector z such that Gz = b. Now Az = FGz = Fb E Z,(A), and there 
is an integral basic solution x0 such that Ax, = FGx, = Fb. Since F has full 
column rank, this implies that Gx, = b. Now the nonzero entries of x0 
correspond to linearly independent columns of A, which correspond to 
linearly independent columns of G. Hence x0 is a basic solution to Gx = b, 
and G E WBSP. Conversely, if G E WBSP and b = AZ E Z,(A) where z is 
integral, then Gz E Z,(G). Hence there is an integral basic solution x0 such 
that Gx, = Gz and Ax, = FGx, = FGz = b. Thus A E WBSP. 
For part (ii) note that the row space of G equals the row space of A, 
which is also given as the row space of any submatrix A, of T linearly 
independent rows of A. Hence there is a nonsingular matrix T such that 
TG = A,. Now A E DAN if and only if A, E DAN [6, (4.3)] if and only if 
G E DAN [6, (4.2)]. 
Part (iii) is given in [2, Theorem 21. 
INTEGRAL MATRICES 479 
For part (iv) suppose that A E BSP. Then A E INT, and hence F E INT 
by (iii). Also A E BSP implies G E BSP by an argument similar to (i). 
Conversely, suppose that G E BSP and F E INT. Then G E INT, and thus 
A E INT by (iii). Also A E WBSP by (‘) I , so that A E BSP by Theorem l(i). 
For part (v) note that there is a correspondence between bases B of A and 
bases C of G such that B = FC. Also there is a correspondence between r X r 
submatrices Bi of B and T x r submatrices Fi of F such that Bi = F,C. Thus 
gcd,(det Bi} = gcd,{det <det C} = (det C)gcdi{det Fi} = det C whenever F E 
INT. Now A E UNM implies F E INT by (iii) and thus G E UNM. Con- 
versely, when F E INT the above identity for determinants holds and hence 
G E UNM implies A E UNM. 
Observe that any r x r submatrix of A is obtained by multiplying r X r 
submatrices of F and G. Thus part (vi) follows, since the determinant of a 
product is the product of determinants, and all subdeterminants of F and G 
are integers. Finally since there is a correspondence between bases of A and 
bases of G, part (vii) is obtained in a manner similar to part (vi). H 
Proof of Theorem l(C). That SUM c SUB n DAN is clear. Let A E SUB 
n DAN, and let A = FG be an integral full-rank factorization. Then F, G E 
SUB and G E DAN by Theorem 2(n), (vi). Now G contains some T X r 
submatrix with determinant _t 1, and hence all nonzero r x r subdeterminants 
ofGare *l.IfCisabasisforG,thendetC*OandhencedetC=kl,that 
is G E SUM. Also F E SUM, since it has full column rank, and A = FG E SUM 
by Theorem 2(vii). W 
Let 
where P and Q are nonsingular unimodular matrices, and D, is an r x r 
integral nonsingular diagonal matrix (the Smith canonical form will yield such 
a factorization). Then a great deal of information is contained in the first r 
rows of Q. Let PO be the submatrix consisting of the first r columns of P, and 
let Q,, be the submatrix consisting of the first r rows of Q. Then A = FQ,, 
where F = POD, gives an integral full-rank factorization of A. Now the Smith 
canonical form of Q0 is (I, 0), so that Q0 E INT and we obtain A E WBSP a 
Q,EWBSP 0 Q,EBSP and AEDAN e Q,EDAN = Q,EUNM. 
Furthermore if 
480 
where k = m - r, then 
JEAN H. BEVIS AND FRANK J. HALL 
Q 
belongs to INT and has the full-rank factorization 
Thus we obtain 
COROLLARY 1. Given an integral matrix A, there exists a nonsingular 
(not necessarily integral) matrix T with an integral inverse szcch that TA E 
INT, TA E BSP * A E WBSP, and TA E UNM 0 A E DAN. Furthermore, 
if S is any nonsingular matrix with an integral inverse such that SA is 
integral, then SA E BSP * A E WBSP and SA E UNM * A E DAN. 
Proof. All that remains is the “Furthermore” portion. Suppose that 
SA = FG is an integral full-rank factorization of SA. Then A = (S - ’ F)G is an 
integral full-rank factorization of A. Now SA E BSP implies that G E BSP and 
hence A E WBSP. Similarly, SA E UNM implies that G E UNM and hence 
A E DAN. W 
Finally we give a characterization of subunimodular matrices in terms of 
both full-rank factorizations and generalized inverses. 
THEOREM 3. Let A be an m x n integral matrix of rank r. Then A is a 
subunimodular matrix if and only if 
(i) A has an integral fill-rank factorization A = FG where 
PF= lB and GQ=(c CN) 
( 1 
for some permutation matrices P and Q, some r X r m&rices B and C, and 
some integral matrices M and N, respectively. 
(ii) A has an integral {l>inverse. 
Proof. - : Use Equation (1) again to write 
INTEGRAL MATRICES 481 
where P and Q are permutation matrices, M and N are integral, and A, is 
nonsingular unimodular. Hence writing 
(Ir N)Q= 
yields (i). Also (ii) holds since the gcd of all T X r subdeterminants of A is 1. 
=: Assuming (ii), from Theorem 2(iii) we then have that both F and G 
have an integral {l)-inverse. Hence, the gcd of all r X r subdeterminants of F, 
and of G, is 1. 
Now, the gcd of all r X r subdeterminants of 
F=p=(izB) and (f&3) 
are the same. But then 
(-'M :)(A&)=(:) 
and since M is integral it follows from [8, Lemma 7.51 that the gcd of all r X r 
subdeterminants of 
(hi?) and (3 
are the same. Thus, the gcd of all r X r subdeterminants of is 1, and 
hence Z? must be nonsingular unimodular. (Note that r = rankOF = rank B, 
which implies B is nonsingular.) Similarly, C is nonsingular unimodular. The 
proof then follows from Theorem 2(vi). n 
REMARKS. 
(1) The characterization of SUB given in Theorem 3 is similar to a 
characterization given in [4] for nonnegative integral matrices with a non- 
negative integral {l>inverse. It is also shown in [4] that a rank-r nonnegative 
integral matrix is in this latter class if and only if it contains under permuta- 
tion I, as a submatrix, and thus this latter class is contained in SUB. 
(2) It is well known [7, Theorem II.11 that if o is an integral vector whose 
coordinates have a gcd of 1, then there is a nonsingular unimodular matrix 
482 JEAN H. BEVIS AND FRANK J. HALL 
containing v as one of its rows. This result is generalized by: 
PROPOSITION 1. Let G be an r X n integral rank-r matrix. Then G E INT 
if and only if there exists an (n - r) X n integral matrix M such that 
G 
Q=( 1 M 
is rwnsingukzr unimodulur. 
Proof. By [7, Theorem II.21 there is a nonsingular unimodular matrix U 
such that GU = (H 0), where H is an integral upper triangular matrix with 
positive diagonal entries and with each element to the right of the diagonal 
being a least residue modulo the diagonal element on its left. If G E INT, 
then by [8, Lemma 7.51 det H = 1. Thus the diagonal entries of H are l’s and 
H=Z,. It follows that G=(Z, O)U-’ and G is the first r rows of U-‘. 
Conversely, if Q = g 
( 1 
is nonsingular unimodular, then the Smith canonical 
form of G is GQ - ’ = (I, 0) and thus G E INT. n 
(3) We note that in characterizing the classes BSP, SUB, and UNM, the 
full-row-rank case is very important (see Theorem 2 and the remark preceding 
its corollary). It could be helpful to have characterizations for the full-row-rank 
matrices in these classes similar to that given in Proposition 1 for INT. That 
is, what additional conditions would one have to put on Q in Proposition 1 in 
order to replace INT by BSP, SUB, or UNM? 
REFERENCES 
S. Bamett, Ma&ices in Co&ml Theory, Van Nostrand Reinhold, 1971. 
D. R. Batigne, F. J. Hall, and I. J. Katz, Further results on integral generalized 
inverses of integral matrices, Linear and Multilinear Algebra 63X-241 (1978). 
F. J. Hall and I. J. Katz, More on integral generalized inverses of integral 
matrices, Linear and Multilinear Algebra 9:201-209 (1980). 
Nonnegative integral generalized inverses, Linear Algebra AppZ. 
39:23-39 (1981). 
I. Heller, On linear systems with integral valued solutions, Pacific J. Math. 
7:1351-1364 (1957). 
On unimodular sets of vectors, in Recent Advances in Mathematical 
Pzrning (R. L. Graves and P. Wolfe, Eds.), McGraw-Hill, 1963. 
M. Newman, Integral Matrices, Academic, 1972. 
S. Perk, Theory of Matrices, Addison-Wesley, 1952. 
INTEGRAL MATRICES 483 
9 H. J. S. Smith, On systems of linear indeterminate equations and congruences, 
Philos. Trans. Roy. Sot. London 151293-326 (1861). 
10 K. Truemper, Algebraic characterizations of unimodular matrices, SIAM 1. A&. 
Math. 35:328X332 (1978). 
11 K. Truemper and R. Chandrasekaran, Local unimodularity of matrix-vector pairs, 
Linear Algebra Appl. 22:65-78 (1978). 
12 A. F. Veinott and G. B. Dantzig, Integral extreme points, SlAM Reu. 10:371-372 
(1968). 
Received 22 October 1981 


Some classes of integral matrices 

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Some classes of integral matrices

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