In mathematics, an eigenvector of a transformation is a non-null vector whose direction is unchanged by that transformation. The factor by which the magnitude is scaled is called the eigenvalue of that vector. Often, a transformation is completely described by its eigenvalues and eigenvectors. An eigenspace is a set of eigenvectors with a common eigenvalue.
These concepts play a major role in several branches of both pure and applied mathematics � appearing prominently in linear algebra, functional analysis, and even a variety of nonlinear situations.
The German word eigen was first used in this context by Hilbert in 1904 (there was an earlier related usage by Helmholtz). "Eigen" can be translated as "own", "peculiar to", "characteristic" or "individual"�emphasizing how important eigenvalues are to defining the unique nature of a specific transformation. In English mathematical jargon, the closest translation would be "characteristic"; and some older references do use expressions like "characteristic value" and "characteristic vector", or even "Eigenwert", German for eigenvalue. In the past, the standard translation used to be "proper". Today the more distinctive term "eigenvalue" is standard.
Transformations of space�such as translation (or shifting the origin), rotation, reflection, stretching, compression, or any combination of these; other transformations could also be listed�may be visualized by the effect they produce on vectors. Vectors can be visualised as arrows pointing from one point to another.
- Eigenvectors of transformations are vectors which are either left unaffected or simply multiplied by a scale factor after the transformation.
- An eigenvector's eigenvalue is the scale factor that it has been multiplied by.
- An eigenspace is a space consisting of all eigenvectors which have the same eigenvalue, along with the zero(null) vector which itself is not an eigenvector.
- The principal eigenvector of a transformation is the eigenvector with the largest corresponding eigenvalue
- The geometric multiplicity of an eigenvalue is the dimension of the associated eigenspace.
- The spectrum of a transformation on finite dimensional vector spaces is the set of all its eigenvalues.
For instance, an eigenvector of a rotation in three dimensions is a vector located within the axis about which the rotation is performed. The corresponding eigenvalue is 1 and the corresponding eigenspace contains all the vectors parallel to the axis. As this is a one-dimensional space, its geometric multiplicity is one. This is the only eigenvalue of the spectrum (of this rotation) that is a real number.
As the Earth rotates, every arrow pointing outward from the center of the Earth also rotates, except those arrows that lie on the axis of rotation. Consider the transformation of the Earth after one hour of rotation: An arrow from the center of the Earth to the Geographic South Pole would be an eigenvector of this transformation, but an arrow from the center of the Earth to anywhere on the equator would not be an eigenvector. Since the arrow pointing at the pole is not stretched by the rotation of the Earth, its eigenvalue is 1.
Another example is provided by a thin metal sheet expanding uniformly about a fixed point in such a way that the distances from any point of the sheet to the fixed point are doubled. This expansion is a transformation with eigenvalue 2. Every vector from the fixed point to a point on the sheet is an eigenvector, and the eigenspace is the set of all these vectors.
Fig. 2. A standing wave in a rope fixed at its boundaries can be seen as an example of an eigenvector, or more precisely, an eigenfunction of the transformation corresponding to the passage of time. As time passes, the standing wave is scaled but its shape is not modified. In this case the eigenvalue is time dependent.
However, three-dimensional geometric space is not the only vector space. For example, consider a stressed rope fixed at both ends, like the vibrating strings of a string instrument (Fig. 2). The distances of atoms of the vibrating rope from their positions when the rope is at rest can be seen as the components of a vector in a space with as many dimensions as there are atoms in the rope.
Assume the rope is a continuous medium. If one considers the transformation of the rope as time passes, its eigenvectors, or eigenfunctions, are its standing waves�the things that, mediated by the surrounding air, humans can experience as the twang of a bow string or the plink of a guitar. The standing waves correspond to particular oscillations of the rope such that the shape of the rope is scaled by a factor (the eigenvalue) as time passes. Each component of the vector associated with the rope is multiplied by this time-dependent factor. The amplitude (eigenvalues) of the standing waves decrease with time if damping is considered. One can then associate a lifetime with the eigenvector, and relate the concept of an eigenvector to the concept of resonance.