Omaväärtusvektoriks

Omaväärtusvektoriks, or eigenvector in English, is a fundamental concept in linear algebra. An eigenvector of a square matrix is a non-zero vector that, when transformed by the matrix, only changes by a scalar factor. This scalar factor is known as the eigenvalue. Mathematically, if A is a square matrix, v is an eigenvector, and lambda is its corresponding eigenvalue, the relationship is expressed as Av = lambda*v.

Eigenvectors are crucial because they represent directions that are invariant under a linear transformation. They reveal the underlying structure and behavior of the transformation. For example, in principal component analysis (PCA), eigenvectors of the covariance matrix are used to identify the directions of maximum variance in a dataset, thus reducing dimensionality while retaining important information.

The computation of eigenvectors typically involves finding the roots of the characteristic polynomial, which is derived from det(A - lambda*I) = 0, where I is the identity matrix. Once the eigenvalues (lambda) are found, they are substituted back into the equation (A - lambda*I)v = 0 to solve for the corresponding eigenvectors (v). Eigenvectors are not unique; any non-zero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.