autovetor

Autovetor is the term used in many languages, including Portuguese, for what in English is called an eigenvector. An eigenvector of a square matrix A is a nonzero vector v that satisfies Av = λv for some scalar λ, called the eigenvalue. In other words, applying the linear transformation represented by A to v results in a vector that is a scalar multiple of v.

To find eigenvectors, one typically determines the eigenvalues by solving the characteristic equation det(A − λI) = 0. For each eigenvalue λ, the corresponding eigenvectors are the nonzero solutions of (A − λI)v = 0. The set of all eigenvectors associated with λ forms the eigenspace, a subspace of the underlying vector space. A matrix may have several eigenvalues and corresponding eigenvectors, and different eigenvectors can correspond to the same eigenvalue if the eigenvalue has greater than one-dimensional eigenspace.

Key properties include that for real symmetric matrices, eigenvectors corresponding to distinct eigenvalues are orthogonal, and a matrix is diagonalizable if there are enough linearly independent eigenvectors. In that case A can be written as A = QΛQ^{-1} with Λ diagonal and Q whose columns are eigenvectors; if Q is orthogonal, A = QΛQ^T.

Common computational methods include the power iteration, which finds the dominant eigenvector, and the QR algorithm, which computes the full set of eigenvalues and eigenvectors. Autovetores have wide applications in areas such as principal component analysis, stability analysis in dynamical systems, and computer graphics, where directions preserved under transformations are of particular interest.