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UDC 519.61:511-33
TWO-CIRCULANT GOLDEN RATIO MATRICES
N. A. Balonina, Dr. Sc., Tech., Professor, korbendfs@mail.ru A. A. Vostrikova, PhD Tech., Associate Professor, vostricov@mail.ru M. B. Sergeeva, Dr. Sc., Tech., Professor, mbse@mail.ru aDepartment of Computing Systems and Networks Saint-Petersburg State University of Aerospace Instrumentation, 67, B. Morskaya St., 190000, Saint-Petersburg, Russian Federation
Purpose: This paper considers two-level quasi-orthogonal matrices, complementing the Mersenne and Euler matrices belonging to the class of Hadamard type matrices, which were first highlighted by J. Hadamard and V. Belevitch. The goal of this note is to develop a theory of such matrices based on preliminary research results. The definitions are provided. Methods: Extreme solutions (using the determinant) have been established by minimization of the absolute values of the elements of the matrices followed their subsequent classification. Results: We give the definitions of a section and a layer of quasi-orthogonal matrices. An example of continuous matrices with varying levels is used to show that the branch of golden ratio matrices is closely associated with the Hadamard and Belevitch matrices. Commentary on the applied aspects of the two-circulant golden ratio matrices and illustrations for some elementary and some interesting cases of Fermat, Mersenne and Euler matrices are provided. Practical relevance: Web addresses are given for other illustrations and other matrices with similar properties. Algorithms to construct golden ratio matrices have been implemented in developing software of the research program-complex.
Keywords — Quasi-Orthogonal Matrices, Hadamard Matrices, Belevitch Matrices, Mersenne Matrices, Euler Matrices, Golden Section, Golden Ratio Matrices.
Introduction
An analysis of Mersenne matrix existence [1, 2] has raised the question of quasi-orthogonal matrices belonging to the Hadamard type matrix family [3], the special cases of which are Belevitch (C-matrices) [4, 5], Hadamard [6, 7], Mersenne [8, 9], Euler [10], and Fermat matrices [11]. The matrices are listed in descending order for n = 4k - d, where d = 0, 1, 2, 3, k > 0 is integer.
A quasi-orthogonal matrix, of order n, is a square matrix A, with ja^j < 1 in each column (and row), with maximum modulus 1, has ATA = ra(n)I, with I the identity matrix and ra(n) is the weight.
The values of the entries we will call "levels" of the matrix. An Hadamard matrix with entries {1, -1} is a two-level matrix. A Mersenne matrix with entries {1, -b}, 0 < b < 1 is also a two-level matrix. Now matrices, themselves, can belong to a layer.
Definition 1. In this paper a matrix layer is a set of quasi-orthogonal matrices with known functions for entries describing their dependence on n = 4k - d for some d and all possible k > 0.
A Mersenne matrix, of order n, has negative entries -b, described by some function b = f(n) and determined for all orders n = 4k - 1. Any fixed (non-varying) Mersenne matrix belongs to this layer. In the same way, Hadamard and Euler matrices with sizes n = 4k - d, d = 0, 2, as described in [1-3],
■ Fig. 2. Hadamard matrix H16 before (a) and after (b) normalization
Fig. 3. Mersenne matrix M15 (a) and histogram of moduli of its elements (b)
■
■ Fig. 4. Normalized Mersenne Mj
belong to layers. Fermat matrices [11] do not form such layer, as their level functions are defined within a narrow set of values n = 2k + 1 for even and some odd values of integer k.
Definition 2. In this paper a section is a set of quasi-orthogonal matrices of different layers, which depend on n = 4k - d for some k and all possible d = 0, 1, 2, 3.
A particular (wider) section can be expanded by Fermat matrices using the same principle — the Fermat matrix (Fig. 1) with size 4k + 1 can be used to find the corresponding Hada-mard matrix (Fig. 2) with size 4k (the main order of the section). The Hadamard matrix can then be used to find a Mersenne matrix (Fig. 3, 4) with size 4k - 1. This last matrix can be used to find an Euler (Fig. 5) matrix with size 4k - 2 [3].
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■ Fig. 5. Euler matrix E14 (a) and histogram of moduli of its elements (b)
Matrices with Few Levels
The matrices mentioned above are the manifestation of a mathematical object, described by its layers and sections [3]. The existence of any matrix in a section entails the existence of all other matrices of the same section because these matrices are mutually dependent.
Besides Hadamard matrices with entries {1, -1} and similar to them Mersenne matrices with entries {1, -b}, 0 < b < 1, there are other matrices with small numbers of levels. The Euler matrix En [3, 11] (shown in Fig. 5) is a square matrix of order n = 4k - 2
with entries {1, -1, b, -b}, where E^E^ = § In,
In is an identity matrix, £ =
(n + 2) + (n - 2)b2 2
and b =1, when n = 6, in other cases ь = q—
2 q — 8
where q = n + 2.
The number of levels is an important characteristic of a matrix set. The low number of levels does not guarantee existence of Belevitch matrices (conference matrices) [4, 5], they do not exist for order n = 4k - 2, if n - 1 is not sum of two squares.
The number of matrix levels increases with the value d in the interval n = 4k - d. Hada-mard matrices have single level (by modulus of elements) matrices as the elements are 1 or -1. Mersenne matrices are two-level matrices; Euler matrices are four-level matrices. All these matrices have some minimal number of levels guaranteeing their existence for pre chosen orders [3].
Many sets of quasi-orthogonal matrices with low numbers of levels do not belong to a layer, they
are special orthogonal per columns (Hadamard type) matrices: conference matrices with three levels of entries {0, 1, -1} are defined for orders shared with the bigger family of four levels Euler matrices. Paley [7] noted that any Hadamard matrix (or quasi-orthogonal matrix respectively) can be used to give matrix of the double size using the Sylvester algorithm. These we call these Sylvester constructions.
In this case, a new matrix branch appears: it does not intersect with any of the previous branches. Paley's observation induces us to study artifact matrices from the orthogonal matrix family (the Hadamard family), including the two-circulant golden ratio matrix. This is considered in this paper. The golden ratio matrix [13] and the two-circulant golden ratio matrix of order 10 lead to G-matrices of orders n = 10 ■ 2k, these sizes 10, 20, 40, 80, 160, 320, 640, etc. hold a special place in image processing algorithms.
Continuous Matrices
Continuous matrices are different from previously observed section matrices of the orthogonal (Hada-mard) family, their level functions depend on more than one argument n. Therefore, for each n they generate not one, but a continuum of quasiorthogonal matrices, described by a parametric dependence. This possibility follows from the interpretation of orthogonal or quasi-orthogonal matrix as a table of vector projections of the required orthogonal basis. We use optimal to denote matrices with maximal determinant. This allows us to get non-varying matrices for this continuum, known as orthogonal (Hadamard) matrices [6, 7].
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Sub-optimal solutions are known for M-matrices The upper module level from set of levels
[14, 15] with a small number of levels. Fig. 6 shows a > b > c > g is a = 1. The second and the third
a continuous M-matrix M10. levels depend on the lower level g as b2 + 2(b - 1) +
g a —c —c a —a a h h a + 2(g - c) + c2 = 0, c = 1/(g + 1). The coloured matrix portrait represents the
a g a —c —c a —a a h h structure and levels of entries — every level has its
—c a g a —c h a —a a h own colour.
—c —c a g a h h a —a a The continuous matrix M10 is a matrix with
h a low number of changeable levels and is notable
M10 = a —c —c a g a h a —a by its solutions: two bounds (Fig. 7, 8) of a conti-
—a a h h a —g —a c c —a nuum.
a —a a h h —a — g —a c c One solution (see Fig. 7) is the two-circulant
h a —a a h h —a ri —a Belevitch matrix C10 since when b = c = a = 1 we
h c ë c have g = 0.
h h a —a a c c —a —g —a We call the second solution (see Fig. 8),
a h h a —a —a c c —a —g with b = c = g < a = 1, as two-circulant golden ratio matrix G10.
Fig. 6. Portrait of matrix M10 (a) and histogram of moduli of its elements (b)
■ Fig. 7. Belevitch matrix C10 (a) and histogram of moduli of its elements (b)
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Fig. 9. Euler matrix E10 (a) and histogram of modules of its elements (b)
■
G10 =
g a
a g
-g a
-g -g
a -g
-a a
a -a
g a
-g -g a
a -g -g g a -g a g a -gag
-a a
-g g
a g g a a -g
a
-a
a g
g
-a
a g g -a -
-a a g g
a -a a g
g a -a -a
-a -g
g g
-a a
-a a a -a g -a
-a
-a
-a -g
-a
-a -g
It is distinguished by the equation g2 + g - 1 = 0, well known by its irrational roots called the golden ratio in the Fibonacci number theory. In this case we are interested in the lower level g = 0.618...
that proportional to the inversion of 1.618... . Matrices with such elements were for the first time provided in [13].
Let's note that histograms for golden ratio matrix G10 and Euler matrix E10 are similar to each other (see Fig. 8, 9), they both are two-level (by modulus) matrices.
Golden ratio matrices
Consequently, all golden ratio matrices are defined on orders n = 10 • 2k.
For them, as for all Hadamard family matrices, matrix G10 is the starting point for the sequence of matrices, found by iterations
G2n =
Gn Gn
Gn -Gn
a
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■ Fig. 10. Hadamard matrix H20 (a) and golden ratio matrix G20 (b)
The value of modulus level g is constant. This implies, that golden ratio matrices and Hadamard-type matrices are two boundary solutions of a continuum matrix (Fig. 10).
Acknowledgements
The authors would like to acknowledge the great help of Professor Jennifer Seberry with translation and discussion of the content of this paper. The authors also wish to sincerely thank Tamara Balonina for converting this paper into printing format.
Conclusion
This paper describes a golden ratio matrix G10 and sequence of such G-matrices, represented by the example G20. These matrices are closely associated with Belevitch and Hadamard-type matrices, their specific structures and algorithms to find them.
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4. Belevitch V. Theorem of 2n-terminal Networks with Application to Conference Telephony. Electr. Commun., 1950, vol. 26, pp. 231-244.
Golden ratio matrices, represented by G10, connect with Belevitch matrices as bounds of continuum. The latter (coexisting with Euler matrices) have no solutions for the orders 22, 34, 58 and so on. So golden ratio matrices do not have a layer by the determination.
The range of application of mathematical models as orthogonal bases is wide [3]. There is a curious idea to use the continuous matrix as a model of phase transformations taking place during the crystallization of cooled alloys [16].
The special boundary points of the continuous matrix can explain the patterns, observed in the tests. The two level golden ratio matrix can be a model reflecting the details of crystal structure [17]. The peculiarities of the quasi-crystal problem are present here — the dichotomy of elements, associated with the golden ratio level and specific orders [18]. Matrix models may be calculated and used to predict the existence of new materials [19].
5. Balonin N. A., Seberry J. A Review and New Symmetric Conference Matrices. Informatsionno-upravli-aiushchie sistemy, 2014, no. 4(71), pp. 2-7.
6. Hadamard J. Résolution d'une Question Relative aux De terminants. Bulletin des Sciences Mathématiques, 1893, vol. 17, pp. 240-246 (In French).
7. Paley R. E. A. C. On Orthogonal Matrices. Journal of Mathematics and Physics, 1933, vol. 12, pp. 311-320.
8. Balonin N. A., Sergeev M. B. M-matrices. Informat-sionno-upravliaiushchie sistemy, 2011, no. 1, pp. 1421 (In Russian).
9. Balonin N. A., Sergeev M. B., Mironovsky L. A. Calculation of Hadamard-Mersenne Matrices. Informat-sionno-upravliaiushchie sistemy, 2012, no. 5(60), pp. 92-94 (In Russian).
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10. Balonin N. A. Existence of Mersenne Matrices of 11th and 19th Orders. Informatsionno-upravliaiushchie sistemy, 2013, no. 2, pp. 89 - 90 (In Russian).
11. Balonin N. A., Sergeev M. В. Two Ways to Construct Hadamard-Euler Matrices. Informatsionno-upravliai-ushchie sistemy, 2013, no. 1(62), pp. 7-10 (In Rusian).
12. Balonin N. A., Sergeev M. В., Mironovsky L. А. Calculation of Hadamard-Fermat Matrices. Informa-tsionno-upravliaiushchie sistemy, 2012, no. 6(61), pp. 90-93 (In Russian)
13. Balonin N. A., Sergeev M. В. Matrix of Golden Ratio G10. Informatsionno-upravliaiushchie sistemy, 2013, no. 6(67), pp. 2-5 (In Russian).
14. Balonin N. A., Sergeev M. В. M-matrices. Informat-sionno-upravliaiushchie sistemy, 2011, no. 1, pp. 1421 (In Russian).
15. Balonin Yu. N., Sergeev M. B. M-matrix of 22nd order. Informatsionno-upravliaiushchie sistemy, 2011, no. 5(54), pp. 87-90 (In Russian).
16. Balonin N. A., Sergeev M. B. M-matrices and Crystal Structures. Vestnik Magnitogorskogo gosudarstven-nogo tekhnicheskogo universiteta, 2013, no. 3(43), pp. 58-62 (In Russian).
17. Shechtman D., Blech I., Gratias D., Cahn J. W. Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Physical Review Letters, 1984, vol. 53, pp. 1951-1954.
18. Комаrоv S. М. Doubt Crystal. Available at http:// elementy.ru/lib/431491 (accessed 30.09.2014) (In Russian).
19. Balonin Yu. N., Sergeev M. B. The Algorithm and Program for Searching and Studying of M-matrices.
Nauchno-tekhnicheskii Vestnik Informatsionnykh Tekhnologii, Mekhaniki i Optiki,, 2013, no. 3, pp. 8286 (In Russian).
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