Department of Mathematics National Institute of Technology, Warangal HADAMARD MATRICES AND THEIR APPLICATIONS Presented by- Ashish Kumar Gupta MSc (Applied Mathematics) 154902

Outline Introduction Definition Examples Kronecker Product Construction of Hadamard matrices Hadamard Inequality Properties of Hadamard matrices Applications Motivation & Conclusion References

Introduction Hadamard matrices are a class of square matrices first described by James Sylvester (1814-1897) in 1867. He called them anallagmatic pavement. In 1893, French mathematician Jacques Hadamard published a paper investigating the values of determinants of square matrices with entries restricted to the set {−1, 1} .He found that the determinants of these matrices have a maximum value. Very often the most difficult problems can be stated with deceptive simplicity. Although a number of associated ideas have been developed in the search for Hadamard matrices, the very existence of these matrices has extensive consequences in many fields of research, such as optimal design theory, information theory and graph theory.

if and only if H is a Hadamard matrix. Definition- A Hadamard matrix is an n × n matrix H with entries in {−1, 1} such that any two distinct rows or columns of H have inner product 0. A Hadamard matrix is an n × n matrix, with entries from the set {−1, 1} , whose determinant attains an absolute value equal to the upper bound of Hadamard matrix is denoted as Theorem: Let H be an n × n matrix with entries from the set {−1, 1} . Then . =

Example: For some orders, there appear to be many different Hadamard matrices.

Examples:- Let’s look into some examples Examples:- Let’s look into some examples. Case -1 n=1 =[1] or [-1] Case -2 n=2

Case -3 n=3 = Case -4 n=4 or 𝟏 𝟏 𝟏 𝟏 −𝟏 +𝟏 𝟏 +𝟏 +𝟏 or −𝟏 𝟏 𝟏 𝟏 −𝟏 +𝟏 𝟏 +𝟏 −𝟏 Answer :- Not Hadamard matrix . 1) Orthogonality is missing

Theorem: There exists a Hadamard matrix of every order 1,2 and 4n where n N. Case -5 n=5,6 and 2n+1 where n N. Hadamard matrix does not exist.

Some more common examples: n=8,12,16

In order to more easily visualize the Hadamard matrices, let us map the +1 elements as white squares, and the –1 elements as black squares. Figure 1 show normalized Hadamard matrices for the orders of n = 1, 2, 4, 8, 12, and 16 and many more.

In Figure, black squares represent −1s and white squares represent 1s , the n × n Hadamard-Sylvester matrix must have n(n − 1)/2 white squares and n(n + 1)/2 black square.

Construction of Hadamard matrices The symbol denotes direct product of matrices If A is the matrix with typical entry aij, then Theorem: If is a Hadamard matrix of order m and is a Hadamard matrix of order n then is a Hadamard matrix of order mn.

Kronecker Product:- The simplest construction of new Hadamard matrices from old is the Kronecker(or tensor) product. In general, if A=(ai j) and B=(bkl) are matrices of size m×n and p×q respectively, the Kronecker product AB is the mp×nq matrix made up of p×q blocks, where the (i, j) block is ai jB.

Example:- Then in general This is the Sylvester construction

Theorem : The Kronecker product of Hadamard matrices is a Hadamard matrix. Paley Type I Hadamard matrix Theorem : Let q be a power of an odd prime. There exists a Hadamard matrix of order q + 1 if q ≡ 3 (mod 4) and a Hadamard matrix of order 2(q + 1) if q ≡ 1 (mod 4). .

Examples: 1)n=12 12-1 =11 =q 2) n=16 can have 16 Examples: 1)n=12 12-1 =11 =q 2) n=16 can have 16 ? Yes ,selvester theorem 3) n=20 ,24,32 can I handle 20,24,23 yes, 19 , 23 and 31 is prime 4) n=28 (28-1) 27=

Case 5 n=36 If this is same recipe(game)but it is very difficult to arrange

Hadamard's Inequality:- If M is an order n matrix with entries from theset {−1, 1} , then Definition:- A Hadamard matrix is an n n matrix, with entries from the set {−1, 1}, whose determinant attains an absolute value equal to the upper bound of Example:-

Since det(H)= 16= , H is a Hadamard matrix Since det(H)= 16= , H is a Hadamard matrix. Since determinants are difficult to calculate for large matrices, there is a useful theorem to verify that matrices are Hadamard. This is often presented as an alternate definition for Hadamard matrices.

Properties of Hadamard Matrices:- 1) Any two rows or two columns are orthogonal. 2)Every pair of rows or every pair of columns differs in exactly n/2 places. 3)A Hadamard matrix may be transformed into an equivalent Hadamard matrix by any of the following operations: • Interchanging any two rows or any two columns • Multiplying any row or any column by –1 • Matrix transpose • The determinant, |Hn| = is maximal by Hadamard’s theorem on determinants

Applications: Telecommunication and signal processings 1 Applications: Telecommunication and signal processings 1. Error control coding peterson(1960) 2. Used in the1960’s Mariner and Voyager spaces probes

The Code Division Multiple Access(CDMA) Walsh Function + Selvester -Hadamard Matrices =Walsh-Hadamard Sequenes transfer of data in short sequences examples : smart phone

The theory and construction of experimental designs group divisible designs Signal Processing: sequences with low autocorrelation are provided by designs with circulant incidence matrices. Factorial design Orthogonal array Analysis of Statistics Coding Theory: A class of binary codes derived from the rows of a Hadamard matrix are optimal with respect to the Plotkin bound. A particular family of examples (derived from a (16; 6; 2) design) are linear, and were used in the Mariner 9 missions. Such codes enjoy simple (and extremely fast) encryption and decryption algorithms. Cryptography

Motivation Horadam: Are the Hadamard matrices developed from twin prime power difference sets cocyclic? Coding theory is a relatively new field of mathematics that deals with methods for ensuring reliable information exchange. Hadamard conjecture 668 Conclusion: Even though Hadamard matrices are conceptually simple, they have some surprising properties and uses. For weaving, Hadamard matrices provide an interesting design foundation for tie-ups.

References “An Overview of Complex Hadamard” Cubes Rose-Hulman Undergraduate Mathematics Journal, by Ben Lantz, Michael Zowada J. Williamson, “Hadamard’s Determinant Theorem and the Sum of Four Squares.” Duke Math. J., vol. 11, pp. 65-81, 1944. A survey onmodular Hadamard matrices Shalom Eliahou, Michel Kervaire https://www.researchgate.net/publication/3835837 Hadamard Matrices and Their Applications W. D. Wallis and A. Hedayat Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve, and extend access to The Annals of Statistics. A. Hedayat and E. Seiden, “F-square and orthogonal F-square de-sign: A generalization of Latin square and orthogonal Latin squares design”, Ann. Math. Stat., vol. 41, pp. 2035–2044, 1970. An introduction to cocyclic generalised Hadamard matrices(K.J. Horadam Department of Mathematics, R.M.I.T. University, Melbourne VIC 3001, Australia