egenvektorrommene

Egenvektorrommene, also known as eigenspaces, are fundamental concepts in linear algebra, particularly when studying linear transformations represented by matrices. For a given square matrix A, an eigenvector v is a non-zero vector that, when multiplied by A, results in a scalar multiple of itself. This scalar is called the eigenvalue, denoted by lambda. The relationship is expressed as Av = lambda v.

An eigenspace associated with a specific eigenvalue lambda of a matrix A is the set of all eigenvectors corresponding to that eigenvalue, along with the zero vector. This set forms a vector subspace, hence the name "eigenspace". Mathematically, for an eigenvalue lambda, the eigenspace E_lambda is defined as the null space (or kernel) of the matrix (A - lambda I), where I is the identity matrix of the same dimension as A. That is, E_lambda = {v | (A - lambda I)v = 0}.

The zero vector always belongs to any eigenspace because (A - lambda I)0 = 0. However, by definition, eigenvectors themselves must be non-zero. The dimension of an eigenspace is called the geometric multiplicity of the corresponding eigenvalue. Eigenspaces provide crucial information about the behavior of a linear transformation. They represent the directions in space that are preserved (scaled but not rotated or sheared) by the transformation. Understanding eigenspaces is vital for diagonalizing matrices, solving systems of differential equations, and analyzing various phenomena in physics and engineering.