Galerkin coarse matrices and the exact coarse-correction error map

Work over the real field with standard coordinate inner products. Let $A_h\in {\mathbb{R}}^{n\times n}$ be symmetric positive definite (equivalently, positive definite in the Cholesky convention of Hermitian positive-definite matrices and Cholesky factorisation A = LL* with positive diagonal) and let $P\in {\mathbb{R}}^{n\times m}$ have full column rank, where $1\le m\le n$. The map $P$ sends coarse-grid coordinate vectors to fine-grid coordinate vectors; its Euclidean adjoint is the transpose $P^{\mathsf{T}}$ by In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix. Define the Galerkin coarse matrix

$A_H:=P^{\mathsf{T}}{A}_{h}P\mathrm{.}$

Whenever $A_H$ is invertible, define the exact Galerkin coarse-correction error map

$C:=I_n-P{A}_H^{-1}{P}^{\mathsf{T}}{A}_h\mathrm{.}$

This definition is conditional on invertibility of $A_H$. The next theorem proves that condition from positive definiteness of $A_h$ and full column rank of $P$. The formula is in exact real arithmetic; a rediscretized operator is not identified with $A_H$ unless equality is separately shown.