Eigenspacejen

Eigenspacejen is a term used in the field of linear algebra, a branch of mathematics. It refers to the set of all eigenvectors associated with a specific eigenvalue of a linear transformation or matrix. Eigenvectors are non-zero vectors that, when a linear transformation is applied, change at most by a scalar factor. This scalar factor is known as the eigenvalue.

The eigenspace corresponding to an eigenvalue lambda of a matrix A is the set of all vectors v such that Av = lambda * v. This set includes the zero vector and is a subspace of the vector space on which the matrix operates. Eigenspaces are crucial in various applications, including stability analysis in dynamical systems, quantum mechanics, and principal component analysis in data science.

The dimension of an eigenspace is known as the geometric multiplicity of the corresponding eigenvalue. If the geometric multiplicity is equal to the algebraic multiplicity (the multiplicity of the eigenvalue as a root of the characteristic polynomial), the eigenvalue is said to be semisimple. Otherwise, the eigenvalue is defective.

Eigenspaces can be computed using various methods, including the characteristic polynomial, diagonalization, and singular value decomposition. They provide valuable insights into the structure and behavior of linear transformations and matrices.