Eigenarvo

An eigenarvo (eigenvalue) of a square matrix A is a scalar λ for which there exists a nonzero vector v satisfying Av = λv. The vector v is called an eigenarvo-vector or eigenvector corresponding to λ. Intuitively, applying A to v preserves its direction while possibly stretching or shrinking its length.

To find eigenarvot, solve the characteristic equation det(A − λI) = 0. The roots λ are the eigenarvot of A (including their algebraic multiplicities). For each eigenvalue, the set of nonzero vectors v solving (A − λI)v = 0 forms the eigenspace associated with λ. The dimension of this space is the geometric multiplicity of the eigenvalue and is at most the algebraic multiplicity.

A key property is that eigenarvot are preserved under similarity transformations: if B = P^{-1}AP, then B has exactly the same eigenarvot as A. If A has n linearly independent eigenvectors, then A is diagonalizable: A = PDP^{-1}, where D is a diagonal matrix containing the eigenarvot of A and the columns of P are the corresponding eigenvectors. For real matrices, if A is symmetric (A = A^T), its eigenarvot are real and eigenvectors corresponding to distinct eigenarvot are orthogonal.

Applications of eigenarvot are widespread. They determine stability and long-term behavior in linear dynamical systems and differential equations. In data analysis, the eigenarvot of covariance matrices underpin Principal Component Analysis (PCA). In quantum mechanics, eigenarvot correspond to observable energy levels, and in numerical linear algebra, algorithms like the QR algorithm compute eigenarvot efficiently.