Determinant evaluations for binary circulant matrices

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1 Spec Matrices 014; : Research Article Open Access Christos Kravvaritis* Determinant evaluations for binary circulant matrices Abstract: Determinant formulas for special binary circulant matrices are derived and a new open problem regarding the possible determinant values of these specific circulant matrices is stated The ideas used for the proofs can be utilized to obtain more determinant formulas for other binary circulant matrices, too The superiority of the proposed approach over the standard method for calculating the determinant of a general circulant matrix is demonstrated Keywords: Determinant; binary circulant matrices MSC: 15A15, 65F40 DOI 10478/spma Received May 8, 014; accepted November 1, Introduction The problem of evaluating determinants is important and interesting in several areas of mathematics due to their numerous applications, mostly as tools for solving linear systems of equations, matrix inversion and eigenvalue problems Although determinants are a mathematical notion with long history, it is very difficult in general to derive analytical formulas for the determinant of an arbitrary given matrix, or for subdeterminants (minors) of it Therefore, they are still nowadays an intensively investigated subject by many mathematicians, eg cf [16], [7] and [18] Mainly, there exist the following ways for calculating the determinant of a matrix [15] A possible idea is the condensation method", which allows to evaluate a determinant inductively, if the method works There exists also the identification of factors" method, which is based on the idea of a standard proof for the Vandermonde determinant evaluation Sometimes there can be found one or more differential or difference equations for the matrix of which the determinant is to be evaluated A determinant can be calculated with application of the LU factorization of a matrix And finally, in a general setting, it is always possible to perform row and/or column operations, or apply the famous Laplace expansion producing usually an inductive evaluation of a determinant Obviously, these methods cannot be applied for every arbitrary matrix, ie, the matrix should have a special structure and/or fulfill specific properties The methods listed above are ordered according to the rigor and the extent of the conditions a matrix must satisfy so that a method can be applied to it, from stringent" to less stringent" But when we have matrices of special structure, sometimes it is possible to establish analytical formulas for their determinants by taking into account their properties This happened already eg for Cauchy matrices, Vandermonde matrices, Hadamard matrices [1, 1], weighing matrices [1] and others, cf [] The benefit from analytical formulas is that they usually lead to efficient determinant evaluations with negligible computational cost and sometimes offer useful insight into the structure of a matrix *Corresponding Author: Christos Kravvaritis: Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece, ckrav@mathuoagr 014 Christos Kravvaritis, licensee De Gruyter Open This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 0 License Download Date /8/18 1:00 PM

2 188 Christos Kravvaritis In the present work - motivated by open problems and numerical experiments reported in [0] - we prove determinant formulas for binary circulant matrices with first row of the form [a a a k b b], (11) for k = 1,,, and formulate a new open problem concerning the possible determinant values of these specific circulant matrices For the sake of simplicity and better presentation most of the proofs are presented only for the case a = 1 and b = 1 This specific case illustrates better the proposed technique without any loss of generality and offers a clear overview The results for general values of a, b have been derived similarly An n n matrix A with first row [a 0 a 1 a n1 ] for some a 0, a 1,, a n1 is called circulant if each of its rows is a right cyclic shift of the row above it Hence, it can be denoted by A = a 0 a 1 a a n1 a n1 a 0 a 1 a n a n a n1 a 0 a n a 1 a a a 0 Circulant matrices are very useful in Coding Theory [4], Digital Image Processing [, 19], Physics [1], Theory of Statistical Designs [14] etc They are used both to approximate and explain the behavior of Toeplitz matrices, as it is known that suita- bly chosen sequences of circulant matrices asymptotically approximate sequences of Toeplitz matrices, cf [5] Actually, from a structural point of view, a circulant matrix is a special Toeplitz matrix, where the first element of each row is equal to the last element of its preceding row From an algebraic point of view, the class of circulant matrices contains all matrices that are diagonalized by the Discrete Fourier Transform A direct consequence of this property is that the class of circulant matrices is an algebra, and most importantly, that the product of two circulant matrices is again circulant Circulant matrices also arise when periodic boundary conditions are used in image processing or in elliptic partial differential equations The results of the present work can be used for expressing in a simplified manner the inverse of binary circulant matrices that may appear during a compact Fourier analysis for multigrid methods, which suggests a representation and analysis of the multigrid components by generating functions or block symbols [10, 11] A construction of (0, 1) circulant matrices with maximal determinant and other problems related to binary circulant matrices are discussed in [17] In particular, binary circulant matrices of the form (11) arise in several real-life applications For instance, such binary circulant matrices are generator matrices of specific Hadamard codes, which have good coding properies, cf, eg, [8] Notation The entries of an (1, 1) matrix are denoted by (+, ) By x we designate the element x, x > 0 The rows of an n n matrix are denoted as r 1, r,, r n The following lemma will be used throughout the paper Lemma 11 (Schur determinant formula) [9, p1] Let B = B 1 B B B 4 and B 1 is nonsingular Then det B = det B 1 det(b 4 B B 1 1 B ) (1) Main Results We prove some results concerning the determinants of binary circulant matrices Download Date /8/18 1:00 PM

3 Determinant evaluations for binary circulant matrices 189 Proposition 1 a) The determinant of a circulant matrix A of order n with first row [a a a b] is [(n 1)a + b] (a b) n1 b) The determinant of a circulant matrix A of order n with first row [a a a b b] is [(n ) a + b] (a b) n1, for n odd, and 0, for n even c) The determinant of a circulant matrix A of order n 4 with first row [a a a b b b] is [(n ) a + b] (a b) n1, for n (mod ) and n 1 (mod ), and 0, for n 0 (mod ) Proof a) It is known that a matrix of order n of the form b a a a b a B = a a b has determinant [(n 1)a + b](b a) n1 It is possible to obtain the matrix A from the matrix B by performing the following n 1 successive row permutations: interchange the row r 1 successively with the rows r i, i =,, n Hence, det A = (1) n1 det B = (1) n1 [(n 1)a + b](1) n1 (a b) n1 = [(n 1)a + b](a b) n1 b) As mentioned in the introduction, one can deal with the special case a = 1, b = 1, so the matrix A has the form: A = We replace the rows r,, r n with the rows r r 1,, r n r 1, respectively, and obtain the matrix We replace the row r 1 with r r r n and obtain the matrix A n4 n Download Date /8/18 1:00 PM

4 190 Christos Kravvaritis Expanding the determinant of A along the first row yields where and det A = (1) n n 4 B = C = det B + (1) n+1 n det C, (1) E D 1 F E D F 0 E is a square matrix of order n We discriminate two cases with respect to the possible forms of the inverse E 1 1) For n odd, ie n odd, we have From (1) we have E 1 = Carrying out carefully the necessary computations yields Since the matrix E is lower triangular, we have Equations (), () and (4) imply since n is odd Similar calculations give det B = det E ( F E 1 D 1 ) () F E 1 D 1 = 0 () det E = () n (4) det B = () n = n1, (5) det C = det E (0 F E 1 D ) = () n (0 ) = n1 (6) Finally, from (1), (5) and (6) and considering that row operations preserve the value of a determinant (thus, det A = det A ) we obtain det A = (1) n n 4 ( n1 ) + (1) n+1 n 4 n1 = (n 4) n1 Download Date /8/18 1:00 PM

5 Determinant evaluations for binary circulant matrices 191 ) For n even, ie n even, we have E 1 = Similar calculations like before yield det B = det C = n1 for this case, so we obtain det A = (1) n n 4 n1 + (1) n+1 n 4 n1 = 0, since n even The general result in the enunciation of the proposition can be derived in a similar manner by replacing in the above procedure the entries, and n4 of the matrices A, B, C with ab, ba and b+ a (n), respectively, and the factor 1 4 in the second step of the proof with a (ba) c) Consider the binary circulant matrix A with first row [+ + + ] Following a similar procedure to the proof of statement b) of the Proposition, we can derive the matrix A n6 n6 n Expanding the determinant of the matrix A along the first row yields where det A = (1) n1 n 6 B = det B + (1) n n det C + (1) n+1 n 6 E D 1 F G 1 det D, (7), Download Date /8/18 1:00 PM

6 19 Christos Kravvaritis and C = D = E D F G E D F G We discriminate three cases with respect to the possible forms of the matrix E 1 1) For n (mod ), ie n (mod ), we have E 1 = According to (1) det B = det E det(g 1 F E 1 D 1 ) (8) Carrying out carefully the necessary computations yields G 1 F E 1 1 D 1 = 1 1 and det(g 1 F E 1 D 1 ) = 1 (9) Since the matrix E is lower triangular, we have det E = () n (10) Equations (8), (9) and (10) imply det B = () n 1 (11) Similar calculations give det C = det E det(g F E 1 D ) = 0 (1) Download Date /8/18 1:00 PM

7 Determinant evaluations for binary circulant matrices 19 and det D = det E det(g F E 1 D ) = 0 (1) Finally, from (7), (11), (1) and (1) we obtain det A = (1) n1 n 6 () n 1 = (n 6) n1, which corresponds to the nonzero value in the enunciation of the proposition for the specific case a = 1, b = 1 ) For n 1 (mod ), ie n 1 (mod ), we have E 1 = ) For n 0 (mod ), ie n 0 (mod ), we have E 1 = Working similarly to the first case, we obtain for the two latter cases the results of the enunciation corresponding to the special values of a, b In a similar way as in Proposition 1b), the general result is derived by replacing in the above procedure the entries, and n6 in the matrices A, B, C, D with a b, b a and b + a (n ), respectively, and the factors 1 6 in the second step of the proof with a (ba) Remark 1 Proposition 1a) is already proved in [6] The proof in [6, Proposition 84] performs the appropriate additions and subtractions of rows in order to conclude the result The present demonstration utilizes row permutations for deriving the outcome Nevertheless, for the sake of completeness and in view of the more general results stated in the following parts b) and c) of Proposition 1, it is meaningful to present a new, compact proof Table 1 presents some numerical results on the calculation of the determinant of binary circulant matrices with first row of the special form under consideration The orders of the matrices tested are in the first column The second column indicates the number k of 1s in the first row of the matrix The determinants in the third column were calculated using the formulae proved in this work The results coincide with the values obtained with the respective command in Matlab It is clear that the standard computations involve large complexity while the theoretical results demonstrated here require simple calculations The theoretical results of Proposition 1, accompanied with numerous experiments on the computer, lead to posing the following open problem Download Date /8/18 1:00 PM

8 194 Christos Kravvaritis Table 1: Numerical experiments for the calculation of the determinant of a circulant matrix of order n k + 1 with first row [+ + + ], where appears k times order k determinant e e e e e e+15 Open Problem Determinants of binary circulant matrices The determinant of a circulant matrix A of order n k + 1 with first row [a a a b b b ] k is det A = { 0, for n p (mod k), k[(n k) a k + b] (a b)n1, otherwise In the first case p is 0 or a factor of k (different from 1 and k) or a positive integer less than k (different from 1) that has common divisors with k For example, we have zero determinant for k = 4 if n 0 or (mod 4); for k = 5 if n 0 (mod 5); for k = 6 if n 0 or or or 4 (mod 6); for k = 10 if n 0 or or 4 or 5 or 6 or 8 (mod 10); for k = 15 if n 0 or or 5 or 6 or 9 or 10 or 1 (mod 15) Remark Although the proposed technique, which was adopted in the proofs of Propositions 1b) and 1c), works for proving the case of general k at the beginning, it cannot proceed further than a specific point More precisely, the respective matrices A, A i (i = 1,, k; similar to the matrices B, C, D in Proposition 1c), the matrices E, E 1, D i, G i, F and F E 1 can be specified analytically following a similar procedure But the matrices F E 1 D i, G i F E 1 D i do not follow a specific pattern, which could be standardized as it was done for the cases k =, Thus, the general case of k, which is stated above as Open Problem, is probably not solvable by means of the suggested strategy Download Date /8/18 1:00 PM

9 Determinant evaluations for binary circulant matrices 195 Furthermore, the main idea used in the proofs of Propositions 1b) and 1c) can be utilized for proving more similar results, like the one presented in the following Proposition Proposition The determinant of a circulant matrix A of order n 4 with first row [a a a b a b] is [(n ) a + b] (a b)n1, for n (mod 4) and n 1 (mod 4), 4[(n ) a + b] (a b)n1 for n (mod 4) and 0, for n 0 (mod 4) Proof Consider the matrix A = Carrying out the same first two steps as in the proof of Proposition 1b) leads to the matrix A n4 n Expanding the determinant of A along the first row yields where we have the matrix B = det A = (1) n1 n 4 det B + (1) n+1 n det C, (14) E D 1 F G 1 Download Date /8/18 1:00 PM

10 196 Christos Kravvaritis and the matrix C = E D F G We discriminate four cases with respect to the possible forms of the matrix E 1 1) For n (mod 4), ie, n 1 (mod 4), we have the matrix E 1 = From (1) we have det B = det E det(g 1 F E 1 D 1 ) (15) Carrying out carefully the necessary computations yields G 1 F E D 1 = + and det(g 1 F E 1 D 1 ) = 4 (16) Since the matrix E is lower triangular, we have Equations (15), (16) and (17) imply Similar calculations give Finally, from (14), (18) and (19) we obtain det A = (1) n1 n 4 det E = () n (17) det B = () n 4 (18) det C = det E det(g F E 1 D ) = () n 4 (19) () n 4 + (1) n+1 n 4 () n 4 = (n 4) n1 Calculations similar to the first case, by taking always into account the special form of the matrix E 1 for every case, yield the results given in the enunciation of the Proposition for the other three cases The general results for arbitrary a, b are derived with the same substitutions mentioned at the end of the proof of Proposition 1b) Download Date /8/18 1:00 PM

11 Determinant evaluations for binary circulant matrices 197 Comparison The comparison with the typical method for calculating ge- nerally determinants doesn t come into question at all due to its demanding computational cost In this section we present a comparison of the proposed formulas for computing the determinant of binary circulant matrices with the following given standard method for calculating the determinant of an arbitrary circulant matrix, cf, eg, [6] Theorem 1 Let C be an n n circulant matrix with first row [c 0 c 1 c c n1 ] Let ω 1,, ω n be the n roots of unity The characteristic roots of C are given by λ 1, λ,, λ n, where for each i = 1,,, n λ i = c 0 ω 0 i + c 1 ω i + c ω i + + c n1 ω n1 i It is well known from matrix theory that the product of the characteristic roots of a matrix equals the determinant of a matrix Thus the following corollary holds Corollary 1 The determinant of an n n circulant matrix C with first row [c 0 c 1 c c n1 ] is given by n λ i = i=1 n (c 0 + c 1 ω i + c ω i + + c n1 ω n1 i ) (1) i=1 Remark 1 Extensive numerical experiments for various orders n and values of a, b, k confirm that the open problem that is posed is in agreement with Corollary 1 However, it is impossible to establish a rigorous theoretical connection between them because Corollary 1 involves also the nth roots of unity and there doesn t seem to be any reference to them according to the algebraic techniques adopted in Proposition 1 that led to the formulation of the open problem If in the general scheme of Corollary 1 one considers the special cases that are handled in Proposition 1, ie, circulant matrix with first row given by (11), the general formula (1) takes the form n nk1 n1 a ω j i + b, where ω k, k = 0,, n 1, are the nth roots of unity i=1 j=0 j=nk ω j i Example 1 In order to gain insight into the previously mentioned theoretical results of this section, consider the 4 4 circulant matrix C with first row [+ + + ] The fourth roots of unity are: 1, 1, i, i According to Theorem 1, the eigenvalues of C are: λ i = 1 + ω i + ω i ω i, ie, λ 1 =, λ =, λ = i, λ 4 = i Corollary 1 yields det C = i (i) = 16, which can be also verified with Proposition 1 For the matrix C having first row [+ + ] the eigenvalues are given by λ i = 1 + ω i ω i ω i, ie λ 1 = 0, λ = 0, λ = + i, λ 4 = i Hence, det C = 0 The same result follows immediately from Proposition 1b) Next consider the circulant matrix C of order n = 100 with first row [+ + + ] The respective roots of unity are given by the formula ω k = cos kπ 100 kπ + i sin, k = 0,, 99 () 100 Taking into account Theorem 1 and Corollary 1 and replacing the 100 roots of unity according to (), the determinant of C can be calculated as det C = λ i = (1 + ω i + ω i + + ω n i i=1 i=1 ω n1 i ) = 6115e + 1 Download Date /8/18 1:00 PM

12 198 Christos Kravvaritis In a similar fashion, one can obtain, eg, for n = 800 that det C = 6605e + 4, which is in agreement with the same result obtained using Proposition 1a) All results can be verified with mathematical software, eg, like Matlab It becomes apparent that the larger the order of the matrix gets the more strenuous the computations become Following an analogous procedure one obtains the results for the other forms of first row, too An overview of the numerical experiments is presented in Table 1 Summarizing, the above described standard method for finding the eigenvalues of an n n circulant matrix C with first row [c 0 c 1 c c n1 ] performs the following three steps 1 find the n roots of unity ω i, i = 0, 1,, n 1 calculate the n eigenvalues as: λ i = c 0 + c 1 ω i + c ω i + + c n1 ω n1 i n det C = λ i i=1 Specifically, the aforementioned steps have the following complexity in a floating point number system A theoretical and crucial measure of efficiency is the number of basic floating poimt operations (flops) needed to implement an algorithm (assition, subtraction, multiplication, division) 1 The determination of the n roots of unity according to the definition in () requires n flops The computation of the eigenvalues of C following Theorem 1 needs n flops The computation of the determinant of C as product of its n eigenvalues performs n flops So there is a total count of n flops The proposed analytical formulae, also the formula stated as open problem, require only 7 flops in the nonzero determinant case The zero case can be handled completely only with a simple modulo check Hence, the superiority of the proposed scheme in comparison with standard techniques from a numerical point of view becomes evident It is obvious that the demonstrated results in this paper offer very simple determinant evaluations with straightforward formulas and are more efficient than the standard method for calculating the determinant of a circulant matrix when applied for computing the determinant of a binary circulant matrix The superiority of the proposed method becomes more apparent by considering large values of the order n 4 Conclusions We prove theoretically the results for determinants of binary circulant matrices that were observed only experimentally, cf [0], and formulate the new Open Problem The proposed approach is particularly simple to apply and outperforms the standard procedure for computing the determinant of a general circulant matrix, when applied to binary circulant matrices Subjects currently under research are the further computation of determinants of other circulant matrices, which could lead to a standardization of the theoretical framework for variations like the one given in Proposition, and the possibility of proving the proposed Open Problem, probably by means of different techniques The implementation of ideas, like the ones in the proofs of Propositions 1 from an algorithm development point of view with the notion of symbolic computing is also challenging, since they seem to follow a predictable, standard procedure Acknowledgement: The author would like to thank the referee for the valuable comments that contributed to a substantial improvement of the paper Download Date /8/18 1:00 PM

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