eigenvektora

An eigenvektora is a nonzero vector v that, when a square matrix A acts on it, is only scaled by a scalar λ, called an eigenvalue: Av = λv. Here A is typically a real or complex n×n matrix, and v ≠ 0.

Eigenvektori and eigenvärden are found from the characteristic equation det(A − λI) = 0. The roots λ of this polynomial are the eigenvalues, and for each eigenvalue the corresponding eigenvectors v satisfy (A − λI)v = 0. The set of all such v forms the eigenspace associated with λ, a subspace whose dimension is the geometric multiplicity of λ. The eigenvalues may be repeated; their algebraic multiplicity equals the multiplicity as a root of the characteristic polynomial.

A matrix is diagonalizable if it has n linearly independent eigenvectors, allowing it to be written as A = PDP^{-1}, where D is a diagonal matrix of eigenvalues and P contains the corresponding eigenvectors as columns. For real symmetric matrices, eigenvectors corresponding to distinct eigenvalues are orthogonal and can be chosen to be orthonormal.

Computation typically involves two steps: first determine the eigenvalues by solving det(A − λI) = 0, then find the eigenvectors by solving the linear system (A − λI)v = 0 for each λ. In practice, numerical methods such as power iteration or QR algorithms are used for larger matrices.

Applications span many fields: principal component analysis uses eigenvectors of the covariance matrix, stability analysis and differential equations rely on eigenvectors to describe modes, and Markov chains use eigenvectors of transition matrices to identify steady states.