Hadamard matrices and the hadamard conjecture

1. Hadamard matrices and the hadamard conjecture By: Ashley reynolds

2. Hadamard Matrices • Linear and multilinear algebra; Matrix theory • Special matrices • Hadamard matrices

3. Definitions • Matrix: In mathematics, a matrix (plural matrices) is a rectangular arrayof numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries.

4. Definitions • Square Matrix: A matrix with the same number of rows and columns. • Identity matrix: a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros.

5. Definitions • Transpose of a matrix: • Orthogonal: having the sum of products of corresponding elements in any two rows or any two columns equal to one if the rows or columns are the same and equal to zero otherwise :  having a transpose with which the product equals the identity matrix.

6. Definitions • Determinant: the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well. The determinant provides important information about a matrix of coefficients of a system of linear equations, or about a matrix that corresponds to a linear transformation of a vector space.

7. Hadamard matrices • a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that every two different rows in a Hadamard matrix represent two perpendicular vectors, while in combinatorial terms, it means that every two different rows have matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows.

8. Properties of a hardamard matrix

9. Hadamard conjecture • The most important open question in the theory of Hadamard matrices is that of existence. The Hadamard conjecture proposes that a Hadamard matrix of order 4k exists for every positive integer k. • A generalization of Sylvester’s construction proves that if Hn and Hmare hadamard matrices of order n and m respectively, then Is a hadamard matrix of order nm. This result is used to produce hadamard matrices of higher order once those of smaller orders are known.

10. Hadamard Conjecture • Sylvester's 1867 construction yields Hadamard matrices of order 1, 2, 4, 8, 16, 32, etc. Hadamard matrices of orders 12 and 20 were subsequently constructed by Hadamard (in 1893). • In 1933, Raymond Paley discovered a construction that produces a Hadamard matrix of order q+1 when q is any prime power that is congruent to 3 modulo 4 and that produces a Hadamard matrix of order 2(q+1) when q is a prime power that is congruent to 1 modulo 4.His method uses finite fields. The Hadamard conjecture should probably be attributed to Paley. • The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is 92. A Hadamard matrix of this order was found using a computer by Baumert, Golomb, and Hall in 1962 at JPL. They used a construction, due to Williamson, that has yielded many additional orders. Many other methods for constructing Hadamard matrices are now known.

11. Hadamard conjecture • In 2005, HadiKharaghani and BehruzTayfeh-Rezaie published their construction of a Hadamard matrix of order 428.As a result, the smallest order for which no Hadamard matrix is presently known is 668. • As of 2008, there are 13 multiples of 4 less than or equal to 2000 for which no Hadamard matrix of that order is known. They are: 668, 716, 892, 1004, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964.

12. Resources • http://designtheory.org/library/encyc/topics/had.pdf • http://homepages.math.uic.edu/~leon/mcs425-s08/handouts/Hadamard_codes.pdf • http://www.renyi.hu/conferences/fourier4/Matolcsi.pdf