Portal:Mathematics

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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

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animation showing a right triangle being duplicated and rearranged in a way that illustrates the Pythagorean theorem
animation showing a right triangle being duplicated and rearranged in a way that illustrates the Pythagorean theorem
An animated geometric proof of the Pythagorean theorem, which states that among the three sides of a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, written as a2 + b2 = c2. A large square is formed with area c2, from four identical right triangles with sides a, b and c, fitted around a small central square (of side length ba). Then two rectangles are formed with sides a and b by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas a2 and b2, which together must have the same area as the initial large square. This is a somewhat subtle example of a proof without words.

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In this shear transformation of the Mona Lisa, the central vertical axis (red vector) is unchanged, but the diagonal vector (blue) has changed direction. Hence the red vector is said to be an eigenvector of this particular transformation and the blue vector is not.
Image credit: User:Voyajer

In mathematics, an eigenvector of a transformation is a vector, different from the zero vector, which that transformation simply multiplies by a constant factor, called the eigenvalue of that vector. Often, a transformation is completely described by its eigenvalues and eigenvectors. The eigenspace for a factor is the set of eigenvectors with that factor as eigenvalue, together with the zero vector.

In the specific case of linear algebra, the eigenvalue problem is this: given an n by n matrix A, what nonzero vectors x in exist, such that Ax is a scalar multiple of x?

The scalar multiple is denoted by the Greek letter λ and is called an eigenvalue of the matrix A, while x is called the eigenvector of A corresponding to λ. These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear situations.

It is common to prefix any natural name for the vector with eigen instead of saying eigenvector. For example, eigenfunction if the eigenvector is a function, eigenmode if the eigenvector is a harmonic mode, eigenstate if the eigenvector is a quantum state, and so on. Similarly for the eigenvalue, e.g. eigenfrequency if the eigenvalue is (or determines) a frequency. (Full article...)

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Topics in mathematics

General Foundations Number theory Discrete mathematics


Algebra Analysis Geometry and topology Applied mathematics
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