eigenspaces

Eigenspaces are special directions associated with a matrix or linear transformation. Think of a transformation changing a space—some directions stretch or shrink without changing shape, called eigenvectors. An eigenspace is the collection of all vectors that behave this way for a particular eigenvalue (the factor by which they stretch or shrink). It’s like a set of directions that stay aligned with themselves during the transformation, forming a subspace. Understanding eigenspaces helps us analyze and simplify complex systems, such as in physics, engineering, or data science, by revealing fundamental patterns or invariant directions within the system.