User:Gennaro Prota/Matrix



Informally, a matrix (plural matrices) is a rectangular disposition of entities. If rows and columns of such a disposition are numbered according to a given, arbitrary, convention then any matrix element can be univocally identified by providing, in order, its row index and its column index. This observation naturally leads to a formal definition: a matrix on a set X is a function which associates ordered pairs of integers to elements of X:


 * For any natural number $$k \ge \; 1$$ let us denote with $$\mathbb{N}_k$$ the subset of $$\mathbb{N}$$ described by $$\{x \in \mathbb{N}: 1 \le \; x \le \; k \}$$
 * Let $$X$$ be a set, $$m$$ and $$n$$ two natural numbers $$\ge \; 1$$: then we define matrix with m rows and n columns, and elements in X any function f


 * $$f: \mathbb{N}_m \times \mathbb{N}_n \mapsto X$$

When dealing with matrices, the notations $$x^i_j$$, $$x_{i j}$$ or $$x_{i,j}$$ are generally preferred over the equivalent $$f(i, j)$$.

Applications

 * quantum mechanics

History
Though the term matrix was only introduced in 1850 by James Joseph Sylvester, the concept, if not formally defined in modern terms, dates back to very ancient times. The first known example of usage of matrices appear in fact in a Chinese text dated between 300 BC and 200 AD: Nine Chapters on the Mathematical Art, which uses them to solve systems of equations.


 * Examples
 * 1) A x B
 * 2) B x A