In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matric ...
, an eigenvector () or characteristic vector of a
linear transformation is a nonzero
vector that changes at most by a
scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by
, is the factor by which the eigenvector is scaled.
Geometrically
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, an eigenvector, corresponding to a
real nonzero eigenvalue, points in a direction in which it is
stretched
Stretching is a form of physical exercise in which a specific muscle or tendon (or muscle group) is deliberately flexed or stretched in order to improve the muscle's felt elasticity and achieve comfortable muscle tone. The result is a feeling ...
by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
, the eigenvector is not rotated.
Formal definition
If is a linear transformation from a vector space over a
field into itself and is a
nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as
where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root associated with .
There is a direct correspondence between ''n''-by-''n''
square matrices and linear transformations from an
''n''-dimensional vec