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In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or β1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has 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. The n-dimensional parallelotope spanned...
In mathematics, the Hadamard product (also known as the element-wise product, entrywise product:βch. 5β or Schur product) is a binary operation that takes two matrices of the same dimensions and produces another matrix of the same dimension as the operands, where each element i, j is the product of elements i, j of the original two matrices. It is to be distinguished from the more common matrix product. It is attributed to, and named after, either French mathematician Jacques Hadamard or German Russian mathematician Issai Schur.
The Hadamard product is associative and distributive. Unlike the matrix...
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I work with centrosymmetric matrices and recently have started exploring the question of the existence of centrosymmetric Hadamard matrices.
Definition: An $n \times m$ matrix $A = (a_{i,j})$ is centrosymmetric if $a_{i,j} = a_{n-i+1, m-j+1}$ for each $i=1,\dots,n$, $j=1,\dots,m$.
Denote by $R_n ...
Stallings' folding algorithm (described by Agol) is probably the best way of doing this, but I thought I'd mention an older algorithm which is also useful, due to Whitehead.
Let $\{w_k\}$ be a collection of words in $F_n$. The Whitehead graph is defined as follows. There are $2n$ vertices, labe...
