Eigenvector
Definition
An Eigenvector is a vector that does not change its direction under the associated linear transformation. Specifically, when a linear transformation is applied to it, the vector is only scaled by a scalar factor, which is the eigenvalue associated with it. The term "eigenvector" comes from the German word "eigen", which means "own" or "self". This reflects the fact that eigenvectors are intrinsic to the transformation and the space it operates on.
Mathematical Description
Given a square matrix A, an eigenvector v is a non-zero vector such that when A is multiplied by v, the result is a scalar multiple of v. This relationship can be written as:
- Av = λv
where λ is the eigenvalue corresponding to the eigenvector v. The scalar λ may be any scalar from the field over which the vector space is defined, typically the real or complex numbers.
Properties
Eigenvectors have several important properties that make them useful in a variety of mathematical contexts:
- Eigenvectors corresponding to distinct eigenvalues are linearly independent. This means that they form a basis for the vector space, and any vector in the space can be expressed as a linear combination of the eigenvectors.
- The set of all eigenvectors of a matrix, each paired with its corresponding eigenvalue, is called the eigensystem of the matrix.
- If a matrix is symmetric, its eigenvectors form an orthogonal basis for the vector space.
- The determinant of a matrix is equal to the product of its eigenvalues, and the trace of the matrix is equal to the sum of its eigenvalues.
Applications
Eigenvectors and eigenvalues have wide applications in various fields of science and engineering.
- In Physics, they are used in the study of linear systems of physical states. For instance, in quantum mechanics, the states of an electron in a hydrogen atom are eigenvectors of the hydrogen atom Hamiltonian.
- In Computer Science, they are used in machine learning algorithms, particularly in dimensionality reduction techniques like Principal Component Analysis (PCA).
- In Engineering, they are used in control theory, signal processing, and systems analysis.
- In Economics, they are used in various models to represent stable state transitions in dynamic systems.