Eigendekompozíciója

Eigendekompozíciója, or eigenvalue decomposition, is a factorization of a matrix into a canonical form. It is a fundamental concept in linear algebra with wide-ranging applications in fields such as physics, engineering, and computer science. The eigendekompozíciója of a square matrix A states that A can be written as the product of three matrices: A = V * D * V^-1, where V is a matrix whose columns are the eigenvectors of A, D is a diagonal matrix whose diagonal elements are the corresponding eigenvalues, and V^-1 is the inverse of V.

Eigenvectors are non-zero vectors that, when multiplied by a matrix, result in a scaled version of the original vector. The scaling factor is known as the eigenvalue. In simpler terms, eigenvectors represent directions that are preserved under the linear transformation described by the matrix, and eigenvalues indicate how much these directions are stretched or compressed.

The existence of an eigendekompozíciója is guaranteed for diagonalizable matrices. A matrix is diagonalizable if it has a full set of linearly independent eigenvectors. Not all square matrices are diagonalizable, but those that are can be simplified significantly through this decomposition. The process involves finding the eigenvalues by solving the characteristic equation det(A - λI) = 0, where λ represents the eigenvalues and I is the identity matrix. Once the eigenvalues are found, the corresponding eigenvectors are calculated by solving the system of linear equations (A - λI)v = 0, where v is the eigenvector. The eigendekompozíciója is a powerful tool for understanding the behavior of linear systems and is crucial for algorithms like Principal Component Analysis (PCA) and spectral clustering.