Eigenraum

Eigenraum is a term used in linear algebra, often translated as eigenvector space or eigenspace. It is associated with linear transformations and matrices. For a given linear transformation represented by a matrix A, an eigenraum is the set of all eigenvectors corresponding to a specific eigenvalue, along with the zero vector.

An eigenvector v of a matrix A is a non-zero vector such that when A is multiplied by v, the result is a scalar multiple of v. This relationship is expressed as Av = λv, where λ is the eigenvalue. The eigenraum for a specific eigenvalue λ, denoted as Eλ, is the set of all vectors x such that Ax = λx. This equation can be rewritten as (A - λI)x = 0, where I is the identity matrix.

The set of all solutions to (A - λI)x = 0 forms a vector subspace. This subspace is the eigenraum corresponding to the eigenvalue λ. The zero vector is always included in an eigenraum, as A0 = λ0 is true for any λ. The dimension of an eigenraum is called the geometric multiplicity of the corresponding eigenvalue. If an eigenraum contains only the zero vector, then the eigenvalue is not associated with any non-zero eigenvectors. The concept of eigenräume is fundamental for understanding the behavior of linear transformations and for applications in various fields like quantum mechanics, signal processing, and data analysis.