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27Математические заметки СВФУ Апрель—июнь, 2019. Том 26, №2
UDC 519.61+514.17
PROPERTIES OF (0,1)-MATRICES OF ORDER n HAVING MAXIMAL DETERMINANT M. Nevskii and A. Ukhalov
Abstract. We give some necessary conditions for the maximality of (0, ^-determinant. Let M be a nondegenerate (0, 1)-matrix of order n. Denote by A the matrix of order n +1 which is obtained from M by adding the (n + 1)th row (0, 0,... , 0,1) and the (n + 1)th column consisting of 1's. We prove that if A-1 = (/¿j) then for all
n+1
i = 1,... , n we have |li j | > 2. Moreover, if | det(M)| is equal to the maximal value
j=1
n+1
of a (0, 1)-determinant of order n, then |li j | = 2 for all i = 1,... , n.
j=1
DOI: 10.25587/SVFU.2019.102.31516 Keywords: (0, 1)-matrix with the maximal determinant, simplex, cube, axial diameter.
1. Preliminaries
In this paper, we present some properties of (0,1)-matrices. Our approach to these results lies in the intersection of discrete mathematics, analysis and geometry of convex bodies. We start with necessary geometric definitions.
Assume n G N, Qn := [0,1]n. Let S be a nondegenerate simplex in Rn. Denote by aS the homothetic copy of S with center of homothety at the center of gravity of S and ratio of homothety a. By symbol di(S) we mean the ith axial diameter of S, i.e., the length of the longest segment in S parallel to the ith coordinate axis. The notion of axial diameter of a convex body was introduced by Scott in [1] and [2]. Define a(S) as the minimal a > 0 such that Qn is contained in a translation of of aS.
Let x(j) = .
) xn i
1 < j < n + 1 , be the vertices of S. Consider the
node matrix of this simplex
A
_(i)
Li
„(2)
\x(n+1)
T(l)
(2) xn
x(n+1)
1\
1
1/
We have vol(S') = where A := det(A). Denote by Aj(x) the determinant obtained from A by changing the jth row to the row (xi,... ,xn, 1). Linear polynomials A j(x) := Asatisfy the property A j
, where Sk- is the Kronecker
1
© 2019 M. Nevskii and A. Ukhalov
delta-symbol. Coefficients of Aj form the jth column of A-1. Assume A-1 = (¡¿j), i.e., Aj(x) = ¡i,jx1 + ... + ¡n,jxn + ¡n+1,j• Since each linear polynomial p can be represented in the form
n+1
p(x) ^^ p(x(j) )Aj (x), j=1
we call Aj the basic Lagrange polynomials related to S. By taking p(x) = x1; .. . , xn, 1, we get
n+1 n+1
]TAj (x)xj) = x, ]TAj (x) = 1. (1)
j=1 j=1
Equalities (1) mean that Aj (x) are the barycentric coordinates of x G Rn with respect to S.
The numbers ¡¿j make it possible to calculate both d^S) and a(S). The ¿th axial diameter of S satisfies
1 1 n+1
di(S)
j=1
see [3]. As it was proved in [4], we have the equality
^ Em. (2)
n1
i=1
Being combined together, (2) and (3) imply the simple connection between a(S) and the elements of A-1:
.. n n+1
«(¿o^EEM- (4)
i=1 j=1
If S C Qn, then dj(S) < 1. Applying (3), we obtain for these simplices a(S) > n. Lassak [5] was the first who proved that if a simplex S C Qn has the maximal possible volume, then
d1(S) = ... = dn(S ) = 1. (5)
Another way to get the same property was given in [4], where it was shown that for any simplex in Qn of maximal volume the equality a(S) = n holds. In view of (3) this is equivalent to (5).
By (a, b)-matrix we mean a matrix whose any element is equal to one of the numbers a or b. The values hn and gn are defined as maximal determinants of (0,1) and (-1,1)-matrices of order n respectively. Denote by vn the maximal volume of an n-dimensional simplex contained in Qn. These numbers are connected by equalities gn+1 = 2nhn, hn = n!vn, see, for example, [6, Theorem 2.1].
If a (0,1)-matrix with the maximal determinant of order n is known, it is possible to construct a maximal volume simplex in Qn. Let us enlarge the row set of such a determinant by the row (0,... , 0). Then simplex S with these vertices is contained
in Qn and has the maximal possible volume. Indeed, consider for S the node matrix A. Then
| det(A)| hn
vol(S) =-:-= —- = vn.
n! n!
One can also obtain nonzero vertices of an n-dimensional simplex in Qn with maximal volume in the same way using the columns of a (0,1)-matrix with the maximal determinant of order n. On the other hand, if an n-dimensional simplex S C Qn with the 0-vertex has the maximal possible volume then the coordinates of its nonzero vertices, being written in rows or in columns, form (0,1)-matrices with maximal determinant of order n.
For some results of the authors concerning numerical characteristics of simplices see, for example, [7].
2. Main Result
Theorem. Suppose M is an arbitrary nondegenerate (0,1)-matrix of order n. Let A be the matrix of order n +1 which is obtained from M by adding the (n +1) th row (0, 0,... , 0,1) and the (n + 1)th column consisting of 1 's. Denote A-1 = (Zjj). Then for each i = 1,... , n
n+1
Ei
j=1
_ , I> 2. (6)
=1
If I det(M)| = hn, then for all i = 1,... , n
n+1
E^" I = 2. (7)
j=1
If for some i we have the strong inequality
n+1
ÉMI > 2, (8)
j=1
then I det(M)| < hn.
Proof. Consider the simplex S with the 0-vertex and the rest vertices coinciding in the coordinate form with the rows of M. Obviously, A is the node matrix of S. Since det(A) = det(M) = 0, the simplex S is nondegenerate. The inclusion S C Qn means that di(S) < 1. Being combined with formula (2) for the axial diameters, this gives (6).
Now assume that | det(M)| = hn. Then we have
| det(A)| | det(M)| h
n
vol(S) = ----— = ---—- = —r = vn-
n! n! n!
Consequently, S is a simplex in Qn with the maximal possible volume. All the axial diameters di(S) are equal to 1, see (5). According to (2) we obtain the conditions
Finally, if for some i the strong inequality (8) holds then di(S) < 1. In this case the volume of S is not maximal and | det(M)| < hn. □
The inverse statement to the second part of the theorem is not true. As a simple example, consider the case when M is the identity matrix of order n. Clearly, | det(M)| = hn only for n = 1. But since
A
/1 0 0 1
00 00
1
1
1 1
A-
10 01
00 00
-1\ 1
-1 1
the equalities (7) are fulfilled for all n and i = 1,... , n. The corresponding simplex S is "the corner simplex" in Qn. Nonzero vertices x(1),... ,x(n) of S coincide with the standard basis of Rn and the last vertex x(n+1) is (0,... , 0). Actually, though all the axial diameters di(S) are equal to 1, the volume of S becomes maximal only in trivial case n = 1.
Now let us note that (0,1)-matrices having maximal determinant can be obtained from (-1,1)-matrices with maximal determinant. Let U be an nondegenerate ( — 1,1)-matrix of order n + 1 and T be a (0,1)-matrix of order n. Suppose that these matrices are connected by the following procedure described in [6].
1. Each column of U beginning with —1 we multiply by -1.
2. Each row of the new matrix beginning with —1 we also multiply by —1. Denote by V the matrix of order n + 1 obtained after steps 1 and 2. Both the first column and the first row of this ( — 1,1)-matrix now completely consist of 1's.
3. Denote by W the submatrix of V which stands in rows and columns numbering 2, ... , n +1. Let us change the elements of W replacing 1 by 0 and —1 by 1. We define T to be the resulting matrix of order n.
Applying the argument from [6, Theorem 2.1] we obtain the equality
I det(T)|
| det(U)|
2 n
If | det(U)| = gn+i, then | det(T)| = hn. Conversely, the condition | det(T)| = hn implies | det(U)| = gn+i.
Obviously, the above procedure can be reversed. Starting with a (0,1)-matrix T of order n, the inverse procedure gives the (-1,1)-matrix V of order n +1. Both the first row and the first column of V consist of 1's and | det(V)| = 2n| det(T)|. If | det(T)| = hn, then | det(V)| = gn+i.
i
3. Example for n = 101
In order to make the proposed method more clear, let us consider an example. Namely, utilizing results from the previous section we will show that the biggest
Fig. 1. (-1, 1)-matrix matrix U of order 101. White squares denote —1's, black squares denote 1's
known (by 2003) determinant of order 101 is not maximal. For the sake of compactness, in this section we present matrices as pictures.
The collection of matrices having maximal known determinants of orders 1,..., 119 can be found on the site [8]. Consider the matrix of order 101 from this site. According the information from the page http: //www.indiana.edu/~maxdet/d101.html this matrix was constructed by Orrick and Solomon in 2003. They don't claim that the given (—1, 1)-matrix have the largest possible determinant but only state that its determinant "surpasses the previous record".
The matrix is shown on Fig 1, where white squares represent —1's, while black
Fig. 2. (0, 1)-matrix T of order 100. White squares denote 0's, black squares denote
1's
squares denote 1's. Using the notations of Section 2, we call this matrix U. In this case n = 100. Following the above procedure, let us transform U to (0,1)-matrix T of order 100. The resulting matrix is shown on Fig 2, where white squares represent 0's and black squares denote 1's. Now let us apply our theorem for M = T:
ioi f Tjf = 2.0225... , i= 1,2,3,
EM = | m = 2-0675..., ¿ = 4,
j=1 [ 2, i = 5,... , 100.
For i < 4 strict inequality (8) holds. This yields | det(T)| < h100. Consequently, we
have | det(U)| < g101. Note that
| det(T)| = 2 • 32 • 597 • 79 = 89740816577942302983638241586569761487623964058002457022666931152343750,
| det(U)| = 11376 • 1097. If we consider A as the node matrix of the corresponding simplex S, we will find that some axial diameters of S are less than 1. Indeed, (2) gives
r = 0.9888... , ¿ = 1,2,3, = { f§ = 0.9673... , i = 4,
[ 1, i = 5,... , 100.
This means that S is not a maximal volume simplex in Q1oo. We have considered 294 matrices from the site http://www.indiana.edu/~maxdet/ . The (-1,1)-matrix U of order 101 used above as an example is the only one that gives a (0,1)-matrix satisfying the last statement of our theorem (see (8)).
All the calculations in this section were performed with the use of Wolfram Mathematica in the symbolic mode. Accordingly, we omited most of intermediate results. The corresponding data and programs are available at http: //dx.doi.org/10.17632/sm3x4xrb42.1
REFERENCES
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2. Scott, P.R. Properties of axial diameters. Bull. Austral. Math. Soc. 39(3), 329-333 (1989) doi: 10.1017/S0004972700003233
3. Nevskii, M. V. On a property of n-dimensional simplices. Mat. Zametki 87(4), 580-593 (2010) (in Russian). doi: 10.4213/mzm7698 English translation: Nevskii, M. V. On a property of n-dimensional simplices. Math. Notes 87(3-4), 543-555 (2010) doi:10.1134/S0001434610030326
4. Nevskii, M. Properties of axial diameters of a simplex. Discrete Comput. Geom 46(2), 301-312 (2011) doi:10.1007/s00454-011-9355-7
5. Lassak, M. Parallelotopes of maximum volume in a simplex. Discrete Comput. Geom 21, 449-462 (1999) doi:10.1007/PL00009432
6. Hudelson, M., Klee, V., and Larman, D. Largest j-simplices in d-cubes: some relatives of the Hadamard maximum determinant problem. Linear Algebra Appl. 241-243, 519-598 (1996) doi:10.1016/0024-3795(95)00541-2
7. Nevskii, M., and Ukhalov, A. Perfect simplices in R5. Beitrage zur Algebra und Geometrie / Contributions to Algebra and Geometry 59(3), 501-521 (2018). https://doi.org/10.1007/s13366-018-0386-6
8. Orrick, W.P., and Solomon, B. The Hadamard Maximal Determinant Problem (website), http://www.indiana.edu/~maxdet/ .
Submitted February 28, 2019 Revised May 29, 2019 Accepted June 3, 2019
Mikhail Nevskii, Alexey Ukhalov Department of Mathematics, P.G. Demidov Yaroslavl State University, Sovetskaya str., 14, Yaroslavl, 150003, Russia [email protected], [email protected]
