Eigenvalue decomposition

Eigenvalue decomposition is a mathematical technique used to analyze matrices, representing complex systems like data or transformations. It involves breaking a matrix into its fundamental components: eigenvectors and eigenvalues. Eigenvectors are special directions in space that remain on their line when the matrix transformation is applied; eigenvalues indicate how much these directions stretch or shrink. This decomposition simplifies understanding the matrix’s behavior, making it easier to analyze systems such as patterns in data, physical transformations, or stability. Essentially, it reveals the matrix’s core characteristics by identifying key directions and how they are scaled.