The abstract mathematical concepts of matrices and vectors are central to many scientific problems. Arrays provide a direct semantic link to these concepts. Indeed, whenever a piece of mathematical literature makes reference to a matrix, one can safely think of an array as the software abstraction that represents the matrix. In scientific literature, an expression such as A_ij is typically used to denote the element in the i ^th row and j ^th column of array A. The corresponding expression in NumPy would simply be A[i,j]. For matrix operations, NumPy arrays also support vectorization (details are addressed in Chapter 3 , Using NumPy Arrays), which speeds up execution greatly. Vectorization makes the code more concise, easier to read, and much more akin to mathematical notation. Like matrices, arrays can be multidimensional too. Every element of an array is addressable through a set of integers called indices, and the process of accessing elements of an array with sets of integers is called indexing. This functionality can indeed be implemented without using arrays, but this would be cumbersome and quite unnecessary.

Efficiency can mean a number of things in software. The term may be used to refer to the speed of execution of a program, its data retrieval and storage performance, its memory overhead (the memory consumed when a program is executing), or its overall throughput. NumPy arrays are better than most other data structures with respect to almost all of these characteristics (with a few exceptions such as pandas, DataFrames, or SciPy's sparse matrices, which we shall deal with in later chapters). Since NumPy arrays are statically typed and homogenous, fast mathematical operations can be implemented in compiled languages (the default implementation uses C and Fortran). Efficiency (the availability of fast algorithms working on homogeneous arrays) makes NumPy popular and important.

The NumPy module is a powerhouse of off-the-shelf functionality for mathematical tasks. It adds greatly to Python's ease of development. The following is a brief summary of what the module contains, most of which we shall explore in this book. A far more detailed treatment of the NumPy module is in the definitive Guide to NumPy, Travis Oliphat. The NumPy API is so flexible that it has been adopted extensively by the scientific Python community as the standard API to build scientific applications. Examples of how this standard is applied across scientific disciplines can be found in The NumPy Array: a structure for efficient numerical computation, Van Der Walt, and others: