We apply the Hadamard equivalence to all the binary matrices of the size m×n and study various properties of this equivalence relation and its classes. We propose to use HR-minimal as a representative of each equivalence class, and count and/or estimate the number of HR-minimals of size m×n. Some properties and constructions of HR-minimals are investigated. Especially, we figure that the weight on an HR-minimal's second row plays an important role, and introduce the concept of Quasi-Hadamard matrices (QH matrices). We show that the row vectors of m×n QH matrices form a set of m binary vectors of length n whose maximum pairwise absolute correlation is minimized over all such sets. Some properties, existence, and constructions of Quasi-orthogonal sequences are also discussed. We also give a relation of these with cyclic difference sets. We report lots of exhaustive search results and open problems, one of which is equivalent to the Hadamard conjecture.