Eigenprojection

Eigenprojection refers to a projection operator that isolates the component of a vector or space associated with a given eigenvalue of a linear operator. In spectral theory, eigenprojections (often called spectral or Riesz projections) project onto the corresponding eigenspace or, more generally, onto the generalized eigenspace associated with a particular eigenvalue.

For a linear operator A on a finite-dimensional vector space, the eigenprojections form a set {P_λ} indexed by eigenvalues λ. They satisfy P_λ^2 = P_λ (idempotent), P_λ P_μ = 0 for λ ≠ μ, and sum to the identity on the invariant subspace: ∑_λ P_λ = I. On the range of P_λ, the spectrum of A is {λ}. When A is diagonalizable with a full set of eigenvectors, P_λ is the projection onto the eigenspace E_λ. In the general (not necessarily diagonalizable) case, P_λ projects onto the generalized eigenspace (the primary component) associated with λ.

Two common ways to compute eigenprojections are:

- If A has distinct eigenvalues (diagonalizable): P_λ = ∏_{μ ≠ λ} (A − μI)/(λ − μ). This is a polynomial in A.

- More generally, for isolated eigenvalues, the Riesz projection formula: P_λ = (1/2πi) ∮_C (zI − A)^{-1} dz, where C is a contour enclosing only λ and no other spectrum.

Eigenprojections enable the spectral decomposition A = ∑_λ λ P_λ, facilitating analysis of A’s action, solving differential equations, and understanding stability. The concept connects with the spectral theorem for normal operators and with Jordan form in non-normal cases.