Embedding and extension properties of Hadamard matrices revisited

© 2018 Dimitrios Christou et al., published by De Gruyter. This work is licensed under the Creative Commons AttributionNonCommercial-NoDerivs 4.0 License. Spec. Matrices 2018; 6:155–165 Research Article Open Access Dimitrios Christou, Marilena Mitrouli*, and Jennifer Seberry Embedding and extension properties of Hadamard matrices revisited 10.1515/spma-2018-0012 Received September 13, 2017; accepted March 12, 2018 Abstract:Hadamardmatrices havemany applications in severalmathematical areas due to their special form


Introduction and motivation
From their initial study in the late 19th century until today, Hadamard matrices, named after the French mathematician Jacques Hadamard (1865Hadamard ( -1963, are proven to be very useful in several scienti c areas, such as computational mathematics and physics, coding theory and cryptography, statistics, informatics, and telecommunications with numerous applications [9]. An Hadamard matrix of order n, denoted by Hn, is an n × n matrix with elements + or − and mutually orthogonal rows and columns, i.e., Hn H n = H n Hn = n In (1) where H n denotes the transpose of Hn and In is the identity matrix of order n. Also, a Hadamard matrix is said to be normalized if it has its rst row and column all 1's. Hadamard himself showed that the matrices of this kind have the maximal determinant | det Hn| = n n/ (2) and he observed that such matrices could exist only if n was 1, 2 or a multiple of 4 [8]. Despite the e orts of several mathematicians, Hadamard's observation remains unproven and has formed the basis of one of the great unsolved mathematical problems, referred to as the Hadamard conjecture. However, several methods for the construction of Hadamard matrices have been developed with the oldest one given by the English mathematician J.J. Sylvester in 1867 who proved that there are (± )-matrices of order t for all positive integers t which have the properties of Hadamard matrices. Such matrices are referred to as the Sylvester-Hadamard matrices. Since the construction of an Hadamard matrix of order 428 published by Kharaghani and Tayfeh-Rezaie [10], the smallest order for which no Hadamard matrix is presently known is 668.
Frequently, in several applications it is useful to know if speci c Hadamard matrices are embedded in other Hadamard matrices of higher order, i.e. if an Hadamard matrix of order m is a submatrix of an Hadamard matrix of order n, when m < n. We denote this by Hm ∈ Hn. Regarding the existing embedding properties of the Hadamard matrices [4,13], it is known that H is embedded in Hn for any order n > and Hn is embedded in H n due to the doubling construction [15], which means that the matrix H n = Hn Hn Hn −Hn is always a Hadamard matrix of order n when Hn is a Hadamard matrix of order n. These properties can be expressed as H ∈ Hn and Hn ∈ H n In 1965, Cohn [3] proved that Hn can have a Hadamard submatrix Hm when m ≤ n . Using matrix algebra, Vijayan [17] proved that (n−k)×n row-orthogonal matrices with ± elements can be extended to n×n Hadamard matrices when k ≤ . More recently, Evangelaras et al. [5] used the distance distribution from coding theory to search for normalized Hadamard matrices of order n embedded in normalized Hadamard matrices of order m ≥ n, and Brent [2] generalized Cohn's result to maximal determinant submatrices of Hadamard matrices showing that if Hn has a maximal determinant submatrix M of order m, then m < ( n + ln n) or m ≥ n − . Several other researchers have dealt with this problem in the past employing mostly combinatorial methods and the approaches that have been developed so far are either constructive [12] or employ the Hadamard conjecture [7] providing partially inconclusive results.
In this paper, we examine the conditions under which an Hadamard matrix of order n−k can be embedded in an Hadamard matrix of order n, denoted by H n−k ∈ Hn. The current approach is based on a relation between the minors of Hadamard matrices presented in [16] and, by employing di erential calculus and elementary number theory, rst, we show that H n− / ∈ Hn and H n− / ∈ Hn. Then, for any positive integers n and k multiples of , we proceed with the generalization which is equivalent to Cohn's result in [3]. The above relation (4) was also considered in [2] where it is proven using Szöllősi's result [16] about the minors of a Hadamard matrix and calculus techniques.
In Section 2, we analyze this approach in more depth providing an alternative proof of Szöllősi's result and an analytic description of the steps of the proof of (4), starting from H n− / ∈ Hn. Then, we provide a new number theoretic proof for H n− / ∈ Hn. Finally, in Section 3, we study the problem whether H n−k can exist embedded in Hn when ≤ k < n, and the connection between the order of the matrix H n−k and the values which form the spectrum of the determinant function for (± )-matrices [4,11].

Embedding properties via minors
The current study of the embedding and extension properties of Hadamard matrices is motivated by the results obtained from [16] which lead to a simple relation connecting the minors of a (± )-matrix. The proof of this result was based on the properties of the generalized matrix determinant. A simpli ed proof of the same result has recently been presented by Banica et al. [1]. A more elegant, direct proof employing only the Jacobi identity [6] is given next. Proposition 1. Let Hn an Hadamard matrix of order n ≥ . If M k denotes the absolute value of a k × k minor of Hn where k = d or n − d, then for any < d < n it holds: Proof. An Hadamard matrix of order n can be considered as a block-matrix of the form Consequently, U is invertible and its inverse has the form: Using Jacobi's determinant identity [6] for U, it follows: Since the absolute determinant of a matrix remains invariant under row or column interchange, the last equation holds for any (n − d) × (n − d) and d × d minors of Hn.
The next lemma speci es the values of the determinant of a square (± )-matrix of order n ≤ and gives a more general property for the determinant of order n > . These values will be useful in the following. Lemma 1 ([4]). Let B be an n × n matrix with elements ± . It holds that i) det B is an integer and n− divides det B, ii) when n ≤ , the only possible values for det B are given in Table 1, and they do all occur.
According to Lemma 1-(i), if M k denotes the absolute value of a k × k minor of a (± )-matrix of order n ≥ k, where p is either a positive integer, or zero.
De nition 1 ([11]). The spectrum of the determinant function for (± )-matrices is de ned to be the set of values taken by p = −k | det R k | as the matrix R k ranges over all k × k (± )-matrices.
Orrick and Solomon give a list of values for p in [11]. They instance all values for k = , , . . . , , and . Also, conjectures have been formulated for k = , , , , and .

. Embeddability of Hadamard matrices of order n −
Considering the above results, we begin the study of the embedding of Hadamard matrices of order n−k when n > and k = . The following proposition can be established.

Proposition 2. An Hadamard matrix of order n − cannot be embedded in an Hadamard matrix of order n for
Proof. If t = , then n = and H n− = H which does not exist. If t = , then n = and H n− = H ∈ H = Hn which is true according to (3). Therefore, an integer t > must be considered in the following.
Assuming that an Hadamard matrix of order n − can be embedded in an Hadamard matrix of order n, the relation (5) where M is the absolute value of a × minor of Hn. However, it is known that and, according to Lemma 1-(ii), the possible non zero values that M can take are or . Considering both cases, the value of M will be denoted by m in the following.
Combining (8) and (9), it follows that: Since t > , every term in (10) can be divided by the non-zero algebraic expression (t − )(t − ). Then, (10) is transformed into Using the function f (x) = ln(x) x− , the above equation (11) can be expressed in the form: The real function f is well de ned and di erentiable in the interval ( , +∞). Moreover, for every x ∈ ( , +∞) it holds: The latter shows that f is a strictly decreasing function in the interval ( , +∞). Hence, for any t > it holds: Consequently, the left part of the equation (12) is always positive whereas its right part is either negative (for m = ) or zero (for m = ). Therefore, the assumption that was made in the derivation of the equation (8) is invalid for any n = t with integer t > . Thus, H n− cannot be embedded in Hn.

. Embeddability of Hadamard matrices of order n −
In the case of H n−k with k = the possible non-zero values that M k can take on are only two. Conversely, when k = the determinant spectrum [11] includes more than two values which depend on a speci c integer p. Hence, a di erent approach must be followed for H n−k when n > and k = . The next proposition illustrates this approach and its proof is based on elementary number theory. Proposition 3. An Hadamard matrix of order n − cannot be embedded in an Hadamard matrix of order n for any n = t with integer t > .
H n− / ∈ Hn , n > Proof. For t = , no matrix H n− can be determined and an integer t > will be considered in the following.
Assuming that an Hadamard matrix of order n − can be embedded in an Hadamard matrix of order n, the relation (5) where M is the absolute value of a × minor of Hn. Moreover, it is known that and, according to (6), M = p · , where p is a positive integer. For the × case it has been con rmed that the possible existing values for the integer p are , , . . . , , , , and [11]. Combining (15) and (16), it follows that: The above equation (17)  Assuming that p is an integer for any integer t > , the equation (17) is written equivalently in the form of an equality between two integers: The next two cases are considered: The prime factorization of the left-hand side of (18) shows that the least power of two is mt , whereas the prime factorization of the right-hand side of (18) shows that the least power of two is exactly + m+ t− . Therefore, considering that p might also be a power of two, it follows that: The above inequality implies that t ≤ which contradicts the hypothesis for the integer t in this case, i.e. t ≥ = .
ii) t is divisible by a prime integer r ≥ If m ≥ is an integer such that r m is the maximum power of r that divides t, then the prime factorization of the left-hand side of (18) shows that the least power of r is r mt , whereas the prime factorization of the right-hand side of (18) shows that the least power of r is exactly r m . Considering again that p might also be a power of r, it follows that: The above inequality contradicts the general hypothesis for the integer t, i.e. t > .
Consequently, there is no integer t > which can give a valid p for the minor M . Therefore, the assumption that was made in the derivation of the equation (15) is invalid for any n = t with integer t > and as a result H n− cannot be embedded in Hn.

Remark 1.
The problem of the non-existence of integers p satisfying (17) can also be investigated using tools from calculus. Speci cally, if p in (17) is regarded as a real function of t, then p(t) is di erentiable in the interval ( , +∞) with rst derivative: It can easily be proven that dp dt > for every t > , which implies that p(t) is a strictly increasing function in the interval ( , +∞). As a result, for every t > , it holds which contradicts the fact that p ≤ [11].

Embeddability of Hadamard matrices
In the proof of Proposition 3, the parameter p plays a key role in the study of the embedding properties of Hadamard matrices. Given a positive integer k, by the Hadamard conjecture the absolute value of the maximal determinant of a (± )-matrix of order k is always less than or equal to k k [8]. Therefore, for any minor M k it holds: Ifp is used to denote the maximum value of p, then ( ) implies that The relation (19) forms a necessary condition for the embeddability of Hadamard matrices. Hence, by studying the range of values that p can take, the results obtained from Propositions 2 and 3 can be generalized for an Hadamard matrix of order n − k.

. Embeddability of Hadamard matrices of order n − k
Let n = t and k = r where t, r are positive integers. Generally, < k < n and thus, < r < t. The cases of {t > , r = } and {t > , r = } have been examined in Propositions 2 and 3. The case of {t > , r > ; t > r} will be considered in the following.
Assuming that an Hadamard matrix of order n − k can be embedded in an Hadamard matrix of order n, the relation (5) Furthermore, it holds: If we combine (6), (21) and (22), we get the next important algebraic relation which connects p, and consequently the spectrum of the determinant function, with the order of the matrix H n−k : If t = r, or equivalently n = k, the integer p attains its maximum valuep, thus p =p. Then, H n−k = H k ∈ H k = Hn which holds as mentioned in (3). Therefore, in the following, we shall examine the existence of positive integers t, r satisfying the inequalities: Let θ = r t . Since t > r, it follows that < θ < and < − θ < . Then, Now, for every θ ∈ ( , ) we consider the real function: Using calculus, it can be proven that h(θ) ≥ , if θ ∈ , , and h(θ) < , if θ ∈ , . The graph of the function h(θ) is illustrated in Figure 1.

. Main results on the embeddability -extendability of Hadamard matrices
The preceding analysis provides conclusive results on the embedding problem of Hadamard matrices H n−k which form the next theorem. Theorem 1. An Hadamard matrix of order n − k cannot be embedded in an Hadamard matrix of order n for any positive integers n and k multiples of when k < n . That is Proof. If n > k, then t > r and θ < . Consequently, it is h(θ) > , which implies that the inequality The aforementioned submatrices are not unique. There are several di erent row and column arrangements (i, j) which also satisfy the conditions for (30) to hold. In Table 2, we summarize the con rmed results obtained from (27) and (30) for n = , , . . . , and k = , , . . . , . embedding pattern for all Hadamard matrices. In particular, for k = n it is known that H k ∈ H k and for k > n it is inferred that if a Hadamard matrix of order n has a k × k submatrix with minor p k− and the value of p is speci cally given by (28), then a Hadamard matrix of order n − k may exist embedded in the Hadamard matrix of order n. For orders n ≤ and k ≤ multiples of 4, we noticed that the values of p obtained from (28) also appear in the spectrum of the determinant function given in [11] and this partially veri es the conditions of Conjecture 1.