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ОБРАБОТКА ИНФОРМАЦИИ И УПРАВЛЕНИЕ X
UDC 004.438
REMARKS ON EXTREMAL AND MAXIMUM DETERMINANT MATRICES WITH MODULI OF REAL ENTRIES < 1
N. A. Balonina, Dr. Sc., Tech., Professor, [email protected]
Jennifer Seberryb, PhD, Professor of Computer Science, [email protected] aSaint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaia St., 190000, Saint-Petersburg, Russian Federation
bCentre for Computer Security Research, EIS, University of Wollongong, NSW, 2522, Australia
Purpose: This note discusses quasi-orthogonal matrices which were first highlighted by J. J. Sylvester and later by V. Belevitch, who showed that three level matrices mapped to lossless telephone connections. The goal of this note is to develop a theory of such matrices based on preliminary research results. Methods: Extreme solutions (using the determinant) have been established by minimization of the maximum of the absolute values of the elements of the matrices followed by their subsequent classification. Results: We give the definitions of Balonin-Mironovsky (BM), Balonin-Sergeev (BSM) and Cretan matrices (CM), illustrations for some elementary and some interesting cases, and reveal some new properties of weighing matrices (Balonin-Seberry conjecture). We restrict our attention in this remark to the properties of Cretan matrices depending on their order. Practical relevance: Web addresses are given for other illustrations and other matrices with similar properties. Algorithms to construct Cretan matrices have been implemented in developing software of the research program-complex.
Keywords — Hadamard Matrices, Conference Matrices, Weighing Matrices, Constructions, Balonin-Mironovsky Matrices, Balonin-Sergeev Matrices, Cretan Matrices.
AMS Subject Classification: 05B20; 20B20.
Definition 1. A real square matrix X = (xi;) of order n is called quasi-orthogonal if it satisfies X^X = XXT = cIn, where In is the nxn identity matrix and "T" stands for transposition, c is constant real number. In this and future work we will only use quasi-orthogonal to refer to matrices with real elements, a least one entry in each row and column must be 1. Hadamard matrices are the best known of these matrices with entries from the unit disk [1].
Definition 2. An Hadamard matrix of order n is an nxn matrix with elements 1, -1 such, that HTH = HHT = nIn, where In is the identity matrix.
The Hadamard inequality [2] says, that Hadamard matrices have maximal determinant for the class of matrices with entries from the unit disk (the moduli of the elements is | xij | < 1 by default). Hadamard matrices can only exist for orders 1, 2 and n = 4t, t an integer (the so called Hadamard conjecture).
The class of quasi-orthogonal matrices with maximal determinant and entries from the unit disk may have a very large set of solutions. Different solutions may give the same maximal determinant. Symmetric conference matrices, a particularly important class of 0, ±1 matrices, are the most well known [3].
Definition 3. A symmetric conference matrix, C, is an nxn matrix with elements 0, +1 or -1, satisfying CTC = CCT = (n - 1) In.
Conference matrices can only exist if the number n - 1 is the sum of two squares. Similar to symmetric conference matrices are quasi-orthogonal matrices W = W(2t, 2t - m) of order n = 2t, with elements 0, +1 or -1, satisfying WW = WW1, =
= (2t - m)In. These are called weighing matrices. It has been conjectured [4] that for n = 4t, there exists a W = W(4t, 4t - m) for all integers 0 < m < 4t.
Definition 4. The values of the entries of the quasi-orthogonal matrix, X, are called levels, so Hadamard matrices are two-level matrices and symmetric conference matrices and weighing matrices are three-level matrices. Quasi-orthogonal matrices with maximal determinant of odd orders have been discovered to have a larger number of levels [5].
Definition 5. A Balonin-Mironovsky [5] matrix, An, of order n, is quasi-orthogonal matrix of maximal determinant. In this remark they are called BM matrices.
Conjecture (Balonin, [6, 7]): there are only 5 Balonin-Mironovsky matrices A3, A5, A7, A9, A11 n +1
with
2
L± m, m < 1, levels.
The 2006 paper [5] gave 5 examples of BM matrices. Order 13 was unresolved. During 2006-2011 Balonin and Sergeev carried out many computer experiments to find the absolute maximum of the determinant of A13.
It was speculated [6] that 13 is a critical order for matrices of odd orders with maximal determinant. Starting from this odd order, the number of levels n +1
k >> . An example of a 6-level (by moduli)
matrix of even order was found and called Yura's matrix Y22 [8] (Fig. 1, a). A student Yura Balonin found this rare solution using DOS-MatLab [8, 9]. The matrix levels are captured by the colour of the squares.
Order n = 22 is special, n - 1 is not sum of two squares, and a symmetric three level conference matrix does not exist. The two circulant matrix Y22
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based on the sequences {-f b a -a a a a a -a a -a}, {a a -a -c -a a d a e a -a} has elements with moduli a = 1, b = 0.9802, c = 0.7845, d = 0.6924, e = 0.5299, f = 0.3076. It appears similar to a conference matrix of order 22 because of the small value for f. A non optimal determinant version was also found with f = 0.0055.
It was then discovered that there is a 22x22 matrix W(22,20) constructed using Golay sequences which gave det(W(22,20)) > det(Y22) (Fig. 1, b).
We note that Golay sequences exist which give W(2n, 2n - 2) with determinant (2n - 2)n for orders 4, 6, 10, 18, 22, 34, 42, 54, 66, 82, 102, 106, 130, 162, 258, 262, 322. In the cases 22, 34,66,106,130, 162, 210, 322 there is no corresponding conference matrix [3].
Conjecture I (Balonin-Seberry, 2014): Suppose a W(2n, 2n - 1) does not exist. Suppose a W(2n, 2n - 2) exists. Then the quasi-orthogonal matrix with maximal determinant is constructed using the W(2n, 2n - 2).
Conjecture II (Balonin-Seberry, 2014): Suppose a W(2n, 2n - 1) does not exist. Suppose that W(2n, k) is the weighing matrix with largest k that exists, then W(2n, k) will give a quasi-orthogonal matrix with near maximal determinant.
For order 58 Balonin found [10] a quasi-orthogonal matrix Y58 with only a few levels and determinant 2-1050, the weighing matrices W(58, k), k = 54, 55, 56, 57, do not exist. The weighing matrix W(58,53) has determinant 1050, so conjecture I only applies for W(2n, 2n - 2) matrices (Fig. 2, a, b).
Fig. 1. Yura's matrix Y22 (a) and a weighing matrix W(22,20) (b)
Fig. 2. A low number of levels matrix of order 58 (a) and a weighing matrix W(58,53) (b)
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The absence of a solution with a low number of levels for n > 13, led Balonin and Sergeev to search for and classify quasi-orthogonal matrices with other properties [5, 6, 11-13].
Definition 6. A quasi-orthogonal matrix with extremal or fixed properties: global or local extre-mum of the determinant, saddle points, the minimum number of levels, or matrices with fixed numbers of levels is called a Balonin-Sergeev matrix. They are called here BSM-matrices.
A Balonin-Mironovsky matrix is a Balonin-Sergeev matrix with the absolute maximum determinant. Balonin-Sergeev matrices with fixed numbers of levels were first mentioned during a conference in Crete, so we will call them Cretan matrices (CM-matrix).
Definition 7. A Cretan matrix, X, of order n, which has indeterminate entries, X 1 , X2 , X3 , X4 , «M , xk is said to have k levels.
It satisfies XTX = XXT = ra(n)In, In the identity matrix, ra(n) the weight, that give a number of equations, called the CM-equations, which make X quasi-orthogonal when the variables (indeterminates) are replaced by real elements with moduli | x^ | < 1.
The XTX = XXT have diagonal entries the weight ra(n) and off diagonal entries 0. CM-matrices can be defined by a function ra(n) or functions x1(n), x2(n), x3(n), x4(n), ..., xk(n). We write CM(n; k; ra(n); determinant) as shorthand.
References
1. Seberry J., Yamada M. Hadamard Matrices, Sequences, and Block Designs. Contemporary Design Theory: A Collection of Surveys. J. H. Dinitz and D. R. Stinson eds. John Wiley and Sons, Inc., 1992, pp. 431-560.
2. Hadamard J. Resolution d'une Question Relative aux De terminants. Bulletin des Sciences Mathématiques, 1893, vol. 17, pp. 240-246 (In French).
3. Balonin N. A., Seberry J. A Review and New Symmetric Conference Matrices. Informatsionno-upravli-aiushchie sistemy, 2014, no. 4(71), pp. 2-7.
4. Wallis (Seberry) Jennifer. Orthogonal (0, 1, -1) matrices. Proc. of First Australian Conf. on Combinatorial Mathematics, TUNRA, Newcastle, 1972, pp. 61-84.
5. Balonin N. A., Mironovsky L. A. Hadamard Matrices of Odd Order. Informatsionno-upravliaiushchie siste-my, 2006, no. 3, pp. 46-50 (In Russian).
6. Balonin N. A., Sergeev M. B. M-matrices. Informatsi-onno-upravliaiushchie sistemy, 2011, no. 1, pp. 14-21 (In Russian).
7. Balonin N. A., Sergeev M. B. Local Maximum Determinant Matrices. Informatsionno-upravliaiushchie sistemy, 2014, no. 1(68), pp. 2-15 (In Russian).
8. Balonin Yu. N., Sergeev M. B. M-matrix of 22nd Order. Informatsionno-upravliaiushchie sistemy, 2011, no. 5(54), pp. 87-90 (In Russian).
Notation: When the variable (indeterminate) entries, X1, %2, X 3, X4, .♦♦, Xk occur Sj, ^2, ^3' s4, ..., sk times in each row and column, we write CM(n; k; s1, s2, s3, s4, ..., sk; ra(n); determinant) as shorthand.
A review and questions of existence are discussed in [7, 13, 14].
Balonin and Sergeev concluded [7, 13] that the resolution of the question of the existence of quasiorthogonal matrices and their generalizations discussed here depends on the order [15]:
— for n = 4t, t an integer, at least 2 levels, a, -b, | a | = | b |, are needed;
— for n = 4t - 1, at least 2 levels, a = 1, -b, b < a, are needed;
— for n = 4t - 2, at least 2 levels, a = 1, -b, b < a, are needed for a two block circulant construction;
— for n = 4t - 3, at least 3 levels, a = 1, -b, c, b < a,c < a, are needed.
Definitions and examples of different types of Cretan matrices will be discussed in future papers.
Acknowledgements
The authors wish to sincerely thank Tamara Balonina for converting this note into printing format.
9. Balonin Yu. N., Sergeev M. B. The Algorithm and Program for Searching and Studying of M-matrices. Nauchno-tekhnicheskii vestnik inform atsionnykh tekhnologii, mekhaniki i optiki, 2013, no. 3, pp. 82-86 (In Russian).
10. Balonin N. A. Quasi-Orthogonal Matrix with Maximal Determinant, Order 58. Available at: http:// mathscinet.ru/catalogue/artifact58 (accessed 15 November 2013).
11. Balonin N. A. Existence of Mersenne Matrices of 11th and 19th Orders. Informatsionno-upravliaiushchie sistemy, 2013, no. 2, pp. 89-90 (In Russian).
12. Balonin N. A., Sergeev M. В. Two Ways to Construct Hadamard-Euler Matrices. Informatsionno-upravli-aiushchie sistemy, 2013, 1(62), pp. 7-10 (In Russian).
13. Sergeev A. M. Generalized Mersenne Matrices and Balonin's Conjecture. Automatic Control and Computer Sciences, 2014, vol. 48, no. 4, pp. 214-220.
14. Balonin N. A., Sergeev M. B. On the Issue of Existence of Hadamard and Mersenne Matrices. Infor-matsionno-upravliaiushchie sistemy, 2013, no. 5(66), pp. 2-8 (In Russian).
15. Balonin N. A., Djokovic D. Z., Mironovsky L. A., Seberry Jennifer, Sergeev M. B. Hadamard Type Matrices Catalogue. Available at: http://mathscinet.ru/ catalogue (accessed 25 September 2014).
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