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Complementary primal and dual projections share nonunit spectrum

Statement

Let $V, U_{1}, U_{2}$ be finite-dimensional real vector spaces, let $A : V \to V^{'}$ be symmetric positive definite, and let $L_{i} : V \to U_{i}$ , $T_{i} : U_{i} \to V$ satisfy $L_{i} T_{i} = I_{U_{i}}$ for $i = 1, 2$ and $T_{1} L_{1} + T_{2} L_{2} = I_{V}$ . Then the operators

$G_{1} = (L_{1} A^{- 1} L_{1}^{T}) (T_{1}^{T} A T_{1}), G_{2} = (T_{2}^{T} A T_{2}) (L_{2} A^{- 1} L_{2}^{T})$

have the same eigenvalues, with the same multiplicities, except possibly at the eigenvalue $1$ .

Facts & Assumptions

Given: Finite-dimensional spaces, $A$ symmetric positive definite, and the four displayed left-inverse and complementary-projection identities.

Proof

1.1givenalgebra

The products $G_{1}$ and $G_{2}$ act on $U_{1}$ and $U_{2}^{'}$ , respectively. Set $X = L_{1} A^{- 1} L_{2}^{T} : U_{2}^{'} \to U_{1}$ and $Y = T_{2}^{T} A T_{1} : U_{1} \to U_{2}^{'}$ . By the given hypotheses, $L_{1} T_{1} = I$ , $L_{2} T_{2} = I$ , and $T_{1} L_{1} + T_{2} L_{2} = I$ . Expanding the products gives $G_{1} - I = - X Y$ and $G_{2} - I = - Y X$ . In detail, $G_{1} - I = L_{1} A^{- 1} ((T_{1} L_{1})^{T} - I) A T_{1} = - L_{1} A^{- 1} L_{2}^{T} T_{2}^{T} A T_{1}$ ; also $Y X = T_{2}^{T} A (I - T_{2} L_{2}) A^{- 1} L_{2}^{T} = I - G_{2}$ .

2.1givenstep 1.1algebra

If $u \in U_{1}$ satisfies $G_{1} u = λ u$ with $λ \neq 1$ , then $X Y u = (1 - λ) u$ , so $Y u \neq 0$ . The identity $G_{2} - I = - Y X$ gives $(G_{2} - I) Y u = - Y X Y u = (λ - 1) Y u$ , hence $G_{2} (Y u) = λ Y u$ . For $v \in U_{2}^{'}$ , the reverse map is $v \mapsto - X v / (λ - 1)$ ; the same identities show it is the inverse on the $λ$ -eigenspaces. Thus these eigenspaces are isomorphic.

3.1givenstep 2.1algebra∎

Choose bases of the finite-dimensional spaces and their dual bases. Both operators are diagonalizable: $T_{1}^{T} A T_{1}$ and $L_{1} A^{- 1} L_{1}^{T}$ are positive definite, so, writing $D_{1} = T_{1}^{T} A T_{1}$ , $G_{1}$ is similar to the symmetric positive definite product $D_{1}^{1 / 2} (L_{1} A^{- 1} L_{1}^{T}) D_{1}^{1 / 2}$ ; likewise $G_{2}$ is similar to $D_{2}^{1 / 2} (L_{2} A^{- 1} L_{2}^{T}) D_{2}^{1 / 2}$ , where $D_{2} = T_{2}^{T} A T_{2}$ . The left-inverse identities make $T_{i}$ injective and $L_{i}$ surjective, proving each factor is positive definite. Therefore equal eigenspace dimensions from step 2.1 give equal algebraic multiplicities for every $λ \neq 1$ .

Remarks

Coefficient-scaled averaging and BDDC: for BDDC, $L_{1} = E_{0}$ , $T_{1} = R$ , and $E_{0} R = I_{\hat{W}}$ .

FETI-DP dual system and preconditioner: for FETI-DP, $L_{2} = B$ , $T_{2} = B_{D}^{T}$ , $B B_{D}^{T} = I_{Λ}$ , and $B_{D}^{T} B + R E_{0} = I_{\tilde{W}}$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Jan Mandel and Bedřich Sousedík, BDDC and FETI-DP under Minimalist Assumptions, §4, Lemmas 4.1–4.2 (standard reference, not scraped)
Olof B. Widlund, Introduction to the Design and Theory of Domain Decomposition Algorithms, DD20 Tutorial, pp. 90–93 (standard reference, not scraped)