## Glossary of terms for the maximal determinant problem

• block form [E2,GK]: Let M=RTR be an n×n matrix. A block of size r is an r×r matrix with diagonal elements n and all other elements 3. M is of block form if it consists of such blocks along the diagonal and elements -1 elsewhere.
• equivalence of matrices: Two matrices are equivalent when one can be obtained from the other by a series of permutations and negations of rows and columns.
• excess: The excess of a Hadamard matrix is the sum of its elements.
• Hadamard conjecture [P]: the conjecture that a Hadamard matrix exists for every allowed order.
• Hadamard matrix [Had]: a matrix with entries {-1, +1} such that rows are pairwise orthogonal. The order of a Hadamard matrix must be 1, 2 or 4n with n an integer.
• Kronecker product: If A is a p×q matrix and B is an r×s matrix, then the Kronecker product, A⊗B, is the pr×qs matrix
```a(1,1)B   a(1,2)B   ...   a(1,q)B
.                         .
.         .               .
.                         .
a(p,1)B   a(p,2)B   ...   a(p,q)B
```
where a(i,j) is the element of A in row i and column j. (Also known as the tensor product.)
• Sylvester construction [Sy]: If A and B are Hadamard matrices of orders p and q, then a Hadamard matrix of order pq is formed by taking the Kronecker product A ⊗ B.
• Sylvester matrix [Sy]: Let S0 be the 1 × 1 (Hadamard) matrix (1). A Hadamard matrix, Sn, of order 2n, is obtained by applying the Sylvester construction, Sn=H2 ⊗ Sn-1, where H2 is the Hadamard matrix of order 2:
```   1  1
1 -1.
```

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Page created 26 January 2002.
Last modified 18 June 2003.
Comments: maxdet@indiana.edu