Hadamard matrix

A Hadamard matrix is a square matrix whose entries are +1 or -1, and whose rows are pairwise orthogonal. In other words, any two distinct rows of a Hadamard matrix have dot product 0, and any row of a Hadamard matrix has dot product equal to the number of rows (or columns) of the matrix.

Hadamard matrices have numerous applications in mathematics, computer science, and physics. They are used in coding theory, signal processing, quantum computing, and experimental design, among other areas. One of the most famous applications of Hadamard matrices is in the construction of low-discrepancy sequences, which are used in numerical integration and simulation.

The existence of Hadamard matrices is closely related to the famous Hadamard conjecture, which states that every positive integer multiple of 4 has a Hadamard matrix of that size. This conjecture remains unsolved, although partial results are known.