ominisarvoja

Ominisarvoja, known in English as eigenvalues, are scalars λ associated with a square matrix A that satisfy A v = λ v for some nonzero vector v, called an eigenvector. In other words, λ scales a nonzero vector v unchanged in direction under the linear transformation defined by A.

Ominisarvoja can be found as the roots of the characteristic polynomial det(A − λI) = 0. For an n × n matrix there are n eigenvalues when multiplicities are counted. The sum of the eigenvalues equals the trace of A, and their product equals the determinant of A. Real matrices may have real eigenvalues or complex conjugate pairs as a result of the characteristic equation.

In addition to the eigenvalues, eigenvectors provide the directions that are unchanged up to a scalar multiple. If A is diagonalizable, A = V Λ V⁻¹, where Λ is the diagonal matrix of eigenvalues and V contains the corresponding eigenvectors as columns. For real symmetric matrices, eigenvalues are real and eigenvectors corresponding to different eigenvalues are orthogonal; this is the spectral theorem.

Computation can be done analytically for small matrices by solving the characteristic equation, or numerically for larger ones using methods such as the QR algorithm or power iteration. If A is real, complex eigenvalues occur in conjugate pairs, influencing the system’s behavior even when observations are real.

Applications are widespread: principal component analysis relies on dominant eigenvalues and eigenvectors; in physics and engineering they determine natural frequencies and modes; in dynamical systems they relate to stability, and in computer science the PageRank algorithm uses the dominant eigenvector of a stochastic matrix.