eigenprojector

An eigenprojector, or eigenprojection, is a projection operator associated with a particular eigenvalue of a linear operator. Let A be a linear map on a finite-dimensional vector space V, and let Eλ denote the eigenspace corresponding to eigenvalue λ. A projector Pλ is a linear operator with Pλ² = Pλ, whose image is Eλ and whose kernel is the direct sum of the other eigenspaces, Ker(Pλ) = ⊕μ≠λ Eμ. In this setting, A and Pλ commute and satisfy A Pλ = Pλ A = λ Pλ; thus Pλ acts as the identity on Eλ and annihilates vectors outside Eλ. When A is diagonalizable with a full eigenbasis, V decomposes as a direct sum of the Eμ, and Pλ is the projection onto Eλ along the sum of the other eigenspaces.

In the special case where A itself is a projector (A² = A), A is the eigenprojection onto its image, with eigenvalues 1 on Im(A) and 0 on Ker(A). In functional analysis, spectral projections Pλ are the eigenprojections associated with A’s spectrum, and for self-adjoint operators they correspond to orthogonal projections onto closed subspaces.

The rank of an eigenprojection equals the dimension of its image, which also equals its trace. This provides a way to quantify how much of the space is “captured” by the λ-eigenspace. Eigenprojectors are fundamental in spectral decompositions, enabling the representation of an operator as a sum of projections onto its eigenspaces.

Example: In R³, the projection onto the xy-plane along the z-axis is an eigenprojection onto the subspace spanned by the x and y axes, leaving those components unchanged and sending the z component to zero.