Eigensubruimtes

Eigensubruimtes, also known as eigenspaces in English, are fundamental concepts in linear algebra, particularly in the study of linear transformations and matrices. An eigenspace corresponding to a specific eigenvalue is the set of all eigenvectors associated with that eigenvalue, together with the zero vector. Formally, for a linear transformation \(T: V \to V\) on a vector space \(V\), the eigenspace corresponding to an eigenvalue \(\lambda\) is defined as the null space (kernel) of the matrix \(T - \lambda I\), where \(I\) is the identity transformation.

Eigenspaces are subspaces of the vector space \(V\), meaning they are closed under addition and scalar multiplication. Each eigenspace is characterized by the eigenvalue \(\lambda\) and can be expressed as the solution space to the homogeneous equation \((T - \lambda I) v = 0\). The dimension of an eigenspace is known as the geometric multiplicity of the eigenvalue, which may be less than or equal to the algebraic multiplicity, the multiplicity of \(\lambda\) as a root of the characteristic polynomial.

Eigenspaces play a crucial role in simplifying linear transformations, enabling diagonalization and spectral decomposition. They provide insight into the structure of a linear operator, revealing invariant subspaces and facilitating computations such as matrix powers and exponentials. Understanding eigensubruimtes is essential in various applications, including differential equations, quantum mechanics, and systems theory.