Projeksiooperaatio

Projeksiooperaatio refers to a projection operation in linear algebra and related fields. It is a linear operator P on a vector space V that is idempotent, meaning P squared equals P (P^2 = P). The image Im(P) is the subspace onto which vectors are projected, while the kernel Ker(P) consists of vectors mapped to zero. In many contexts, V decomposes as a direct sum V = Im(P) ⊕ Ker(P), with each vector uniquely written as a sum of a projected component and a component in the kernel.

Key properties include that the rank of P equals the dimension of its image, and the eigenvalues of P are 0 and 1. The 1-eigenspace is Im(P) and the 0-eigenspace is Ker(P). If P is self-adjoint (P = P* with respect to an inner product), the projection is orthogonal, and Im(P) is the orthogonal complement of Ker(P). In general, projections can be oblique, projecting onto Im(P) along Ker(P) without requiring orthogonality.

Examples help illustrate the concept. In R^3, projecting onto a line spanned by a unit vector u has the operator P = u u^T. Projecting onto a coordinate plane, such as the xy-plane, yields P = diag(1, 1, 0). These concrete instances reflect the broader idea of reducing a vector to a subspace.

Applications are widespread. In data analysis and statistics, projections reduce dimensionality and extract features by mapping data onto a subspace. In signal processing, they separate signal from noise. In computer graphics and geometry, projections relate to transforming 3D objects to 2D planes. In numerical linear algebra, projection operators underpin various decomposition and solution techniques. See also projection operator, orthogonal projection, and subspace decomposition.