Hadamard matrix
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 by the rows of an n × n Hadamard matrix has the maximum possible n-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1.
Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determinant problem.
Let H be a Hadamard matrix of order n. The transpose of H is closely related to its inverse.
In fact: where In is the n × n identity matrix and HT is the transpose of H. To see that this is true, notice that the rows of H are all orthogonal vectors over the field of real numbers and each have length
Dividing H through by this length gives an orthogonal matrix whose transpose is thus its inverse.
[1] The proof of the nonexistence of Hadamard matrices with dimensions other than 1, 2, or a multiple of 4 follows: If
Examples of Hadamard matrices were actually first constructed by James Joseph Sylvester in 1867.
The elements in all the other rows and columns are evenly divided between positive and negative.
If we map the elements of the Hadamard matrix using the group homomorphism
, we can describe an alternative construction of Sylvester's Hadamard matrix.
matrix whose columns consist of all n-bit numbers arranged in ascending counting order.
The most important open question in the theory of Hadamard matrices is one of existence.
Sylvester's 1867 construction yields Hadamard matrices of order 1, 2, 4, 8, 16, 32, etc.
[4] In 1933, Raymond Paley discovered the Paley construction, which 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.
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.
In 2005, Hadi Kharaghani and Behruz Tayfeh-Rezaie published their construction of a Hadamard matrix of order 428.
[8] As a result, the smallest order for which no Hadamard matrix is presently known is 668.
entries randomly deleted, then with overwhelming likelihood, one can perfectly recover the original matrix
The algorithm of recovery has the same computational cost as matrix inversion.
[11] Many special cases of Hadamard matrices have been investigated in the mathematical literature.
This makes it possible, for example, to normalize a skew Hadamard matrix so that all elements in the first row equal 1.
This correspondence in reverse produces a doubly regular tournament from a skew Hadamard matrix, assuming the skew Hadamard matrix is normalized so that all elements of the first row equal 1.
A necessary condition on the existence of a regular n × n Hadamard matrix is that n be a square number.
Moreover, if an n × n circulant Hadamard matrix existed with n > 1 then n would necessarily have to be of the form 4u 2 with u odd.
[13][14] The circulant Hadamard matrix conjecture, however, asserts that, apart from the known 1 × 1 and 4 × 4 examples, no such matrices exist.
[16] Another generalization defines a complex Hadamard matrix to be a matrix in which the entries are complex numbers of unit modulus and which satisfies H H* = n In where H* is the conjugate transpose of H. Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation.
The term complex Hadamard matrix has been used by some authors to refer specifically to the case q = 4.
