Hadamard Matrices Abstract Since their discovery in 1893, it has been found that Hadamard matrices have many useful applications. In this paper, we consider some examples of Hadamard matrices and how they are constructed. We also explore three applications in depth; namely factor screening, encrypting and error correcting codes, and briefly discuss some further applications. Max Arnott Jonathan Cound Chlo´eFearn Amiee Rice Julian Villar Emma Wright Lancaster University Supervised by Jan Grabowski Contents 1 An Introduction to Hadamard Matrices 2 1.1 Properties of Hadamard Matrices . .2 1.2 Construction of Hadamard Matrices . .3 1.2.1 Sylvester Matrices using the Kronecker Product . .3 1.2.2 The Paley Construction . .4 2 Applications 7 2.1 Factor Screening . .7 2.2 Encryption . .8 2.3 Error Correcting Codes . 11 1 1 An Introduction to Hadamard Matrices We begin with the definition of a Hadamard matrix. Definition 1.1 ([1]). A Hadamard matrix, H, is a square matrix whose entries must be ±1 and that satisfies HHT = nI; where n is the order of H and I is the n × n identity matrix. We denote 1 as + and −1 as − as follows: 2 1 1 1 1 3 2 + + + + 3 6 1 1 −1 −1 7 6 + + − − 7 H = 6 7 = 6 7 : 4 1 −1 1 −1 5 4 + − + − 5 1 −1 −1 1 + − − + Two trivial examples of Hadamard matrices are the single entry matrices where n = 1: [+] ; [−] : 1.1 Properties of Hadamard Matrices There are many features and properties that define a Hadamard matrix. We discuss these below. Propositionp 1.2 ([2]). The rows and columns of a Hadamard matrix are pairwise orthogonal, with length n. As a direct result of Proposition 1.2, in any Hadamard matrix, if we take two neighbouring rows or columns, half of the adjacent elements have the same sign and half have the opposite sign. Proposition 1.3. Hadamard matrices are restricted so that n = 1; 2 and 4m; m 2 N. This proof is taken from notes made available by the University of Colorado [4]. Proof. Suppose that H is a Hadamard matrix of order n > 2. Then we multiply all columns that have the first entry as − by −1. This will result in the top row containing only + values. Rearrange the first three rows so that they look like: + ::: + + ::: + + ::: + + ::: + + ::: + + ::: + − ::: − − ::: − ::: ::: ::: + ::: + − ::: − + ::: + − ::: − x y z w Here, x, y, z and w are the number of columns of each type. Therefore, x + y + z + w = n: When we take the inner products of rows 1 and 2, 1 and 3, and 2 and 3 we get x + y − z − w = 0 x − y + z − w = 0 x − y − z + w = 0: 2 Solving this system of equations gives x = y = z = w = n=4: Thus, the integer n must be divisible by 4. Remark: Note that this does not prove that a Hadamard matrix exists of order 4m for every positive integer m; it was Jacques Hadamard himself who proposed this conjecture, but it is yet to be proven. Definition 1.4 ([5]). Hadamard matrices are said to be equivalent if they can be obtained from each other by a series of row/column operations which include the negation of row and columns and swapping rows or columns. Hadamard matrices of order 1, 2, 4, 8, and 12 contain a unique Hadamard matrix up to equivalence. There exist 5 non-equivalent Hadamard matrices that have order 16, 3 of order 20 and 60 of order 24. For a long period of time, the smallest order that could not be constructed was 92, but by using computers, mathematicians at JPL [6] have since found matrices of order 4k up to 4k = 668. 1.2 Construction of Hadamard Matrices Hadamard matrices can be constructed in many ways, some of which are unfortunately beyond the scope of this paper. We explore two of the most elementary methods below. 1.2.1 Sylvester Matrices using the Kronecker Product We begin with Sylvester matrices, which are constructed using the Kronecker product of two ma- trices, which we define below. Definition 1.5 ([7]). The Kronecker product, A ⊗ B, of an m × n matrix A and a p × q matrix B is 2 3 a11B ··· a1nB 6 . 7 A ⊗ B = 4 . 5 : am1B ··· amnB Lemma 1.6 ([7]). The following two properties hold for all matrices A; B; C and D: • (A ⊗ B)T = AT ⊗ BT ; and • (A ⊗ B)(C ⊗ D) = AC ⊗ BD. Proofs for the above lemma are omitted, but can be found in notes by Broxson, from the University of North Florida [8]. Theorem 1.7. [[1]] The Kronecker product of Hadamard matrices is a Hadamard matrix. 3 Proof. Let A and B be Hadamard matrices, of size m and n respectively. Then AAT = mI and BBT = nI. Since (A ⊗ B)T = AT ⊗ BT [7], we have, by lemma 1.6 (A ⊗ B)(A ⊗ B)T = (A ⊗ B)(AT ⊗ BT ) = AAT ⊗ BBT = mI ⊗ nI = mnI : So A ⊗ B is a Hadamard matrix of order mn. This gives us the means to construct Hadamard matrices of order st, where matrix A has order s and matrix B has order t. Example 1.8 (The Sylvester Construction). Let + + A = : + − Then to construct a 4 × 4 Hadamard matrix, we can use the Kronecker product of A with itself: 2 + + + + 3 0 AA 6 + − + − 7 A = A ⊗ A = = 6 7 : A −A 4 + + − − 5 + − − + We can check that A0 is a Hadamard matrix by computing A0(A0)T and checking that it is equal to 4I4. Theorem 1.7 gives rise to the following corollary. Corollary 1.9 ([9]). There exist Hadamard matrices of size 2k for all k 2 N. 1.2.2 The Paley Construction A key property of the Paley construction of Hadamard matrices is that we do not require an inital Hadamard matrix to make a new one as we do when using the Sylvester construction; it is not a recursive process. We begin with a definition of a Galois field, recalling from MATH225 [10] the definition of a field. Definition 1.10 ([9]). A Galois field, GF(s), is a field with s elements, where s is finite. Next, we define quadratic residues. Definition 1.11 ([9]). Let GF(s) be a Galois field of order s, where s = pn, for a prime number p and a natural number n. Then an element a in GF(s) is a quadratic residue if and only if there exists b in GF(s) such that a = b2. Otherwise, a is a quadratic non-residue. The quadratic character, χ(a), is defined by Wikipedia [11] as, 8 < 1 if a is a quadratic residue in GF (s); χ(a) = −1 if a is a quadratic non-residue in GF (s); : 0 if a = 0: The Jacobsthal matrix S for GF (s) is an s × s matrix with entries Sab = χ(a − b) [11]. We have two cases: 4 Case 1: s ≡ 1 mod 4, and Case 2: s ≡ 3 mod 4. Let J be an s-dimensional column vector of 1s. Then the next step for constructing the Hadamard matrix is as follows [11]: Case 1: Replace all zero entries in 0 J T JJS with the matrix + − − − and all ±1 entries with the matrix + + ± : + − Then we have a Hadamard matrix of size 2(s + 1). Case 2: 0 J T I + −JJS is a Hadamard matrix of size s + 1. Below, we use the Paley construction to create a Hadamard matrix of order 12. Example 1.12 (Paley Construction). Consider the Galois field Z5 = f0; 1; 2; 3; 4g: Then we have 12 ≡ 1; 22 ≡ 4; 32 ≡ 9 ≡ 4; and 42 ≡ 16 ≡ 1: So 1 and 4 are quadratic residues in Z5, whilst 2 and 3 are quadratic non-residues in Z5. So χ(0) = 0; χ(1) = 1; χ(2) = −1; χ(3) = −1; and χ(4) = 1: Then by computing χ(a − b) for each entry in the Jacobian matrix S, we obtain 0 0 + − − + 1 B + 0 + − − C B C S = B − + 0 + − C : B C @ − − + 0 + A + − − + 0 5 Then, since 5 ≡ 1 mod 4, we can take 0 J T JJS and replace all 0s, 1s, and −1s with + − + + − − ; ; and − − + − − + respectively, which gives 0 + − + + + + + + + + + + 1 B − − + − + − + − + − + − C B C B + + + − + + − − − − + + C B C B + − − − + − − + − + + − C B C B + + + + + − + + − − − − C B C B + − + − − − + − − + − + C H = B C : B + + − − + + + − + + − − C B C B + − − + + − − − + − − + C B C B + + − − − − + + + − + + C B C B + − − + − + + − − − + − C B C @ + + + + − − − − + + + − A + − + − − + − + + − − − We can check that H is Hadamard by multiplying it with its transpose, and we find that HHT = 12I: Sylvester first proposed a method of construction that produced Hadamard matrices of order 2n for all positive integers n. The Paley construction produces Hadamard matrices of order either q + 1 or 2(q + 1) where q is congruent to 3 or 1 mod 4 respectively. 6 2 Applications 2.1 Factor Screening Factor screening is an important application of Hadamard matrices. In factor screening, we investi- gate which factors in an experiment are having the largest effect on the response. We begin by using linear regression which investigates the effects of the explanatory variables on the response variable; this is exactly what we investigate in factor screening. The multiple linear regression model is as follows, as stated in part 2:2 of Jin's work on factor screening [12]: For i = 1; : : : ; n, Yi = β0 + β1xi;1 + β2xi;2 + ::: + βpxi;p + i where the residuals 1; : : : ; n, are independent and identically distributed and follow a normal distribution. So for i = 1; : : : ; n, 2 i ∼ N(0; σ ) : We can also write this as follows [12]: Y = Xββ + 0 0 where Y = (Y1;:::;Yn) is the response vector , X is the design matrix, β = (β1; : : : ; βp) is the 0 vector of coefficients and = (1; : : : ; p) is the residual vector.