
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 $R^{n}$ 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.
Conway's Game of Life is a
cellular automaton devised by the British mathematician
John Horton Conway in 1970. It is an example of a
zeroplayer game, meaning that its evolution is completely determined by its initial state, requiring no further input as the game progresses. After an initial pattern of filledin squares ("live cells") is set up in a twodimensional grid, the fate of each cell (including empty, or "dead", ones) is determined at each step of the game by considering its interaction with its eight nearest
neighbors (the cells that are horizontally, vertically, or diagonally adjacent to it) according to the following rules: (1) any live cell with fewer than two live neighbors dies, as if caused by underpopulation; (2) any live cell with two or three live neighbors lives on to the next generation; (3) any live cell with more than three live neighbors dies, as if by overcrowding; (4) any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction. By repeatedly applying these simple rules, extremely complex patterns can emerge. In this animation, a
breeder (in this instance called a
puffer train, colored red in
the final frame of the animation) leaves
guns (green) in its wake, which in turn "fire out"
gliders (blue). Many more complex patterns are possible. Conway developed his rules as a simplified model of a hypothetical machine that could build copies of itself, a more complicated version of which was discovered by
John von Neumann in the 1940s. Variations on the Game of Life use
different rules for cell birth and death, use more than two states (resulting in evolving multicolored patterns), or are played on a different type of grid (e.g.,
a hexagonal grid or a threedimensional one). After making its first public appearance in the October 1970 issue of
Scientific American, the Game of Life popularized a whole new field of mathematical research called
cellular automata, which has been applied to problems in
cryptography and
errorcorrection coding, and has even been suggested as the basis for new
discrete models of the universe.