High-contrast homogenization: Setup.SchurPositivity #
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Positivity of the Schur block and of the conjugated skew-corrected form #
The reference text reads |σ_*^{-1/2}(σ + (k - h)ᵗ σ_*⁻¹ (k - h))σ_*^{-1/2}|,
the intrinsic contrast of the reference block, as the spectral norm of a
symmetric positive semidefinite matrix. This file supplies that reading: on a
symmetric positive definite doubled block matrix H, the conjugated
skew-corrected form
σ_*^{-1/2}(σ + (k - h)ᵗ σ_*⁻¹ (k - h))σ_*^{-1/2} is symmetric and positive
semidefinite, for every skew parameter h — indeed for every h at all.
The substantive input is the positivity of the Schur block σ. It comes from
the Schur complement identity: the doubled quadratic form of H evaluated at the
state (p, k p) built from the Schur skew block is exactly the quadratic form of
σ at p, so positive definiteness of H transfers to σ.
On real matrices, conjugate transpose agrees with transpose.
Symmetry of the blocks #
The upper-left block of a symmetric doubled block matrix is symmetric.
The off-diagonal blocks of a symmetric doubled block matrix are transposes of one another.
The Schur complement identity #
The Schur complement identity. Evaluated at the state (p, k p) built
from the Schur skew block, the doubled quadratic form of a symmetric doubled
block matrix with invertible lower-right block is the quadratic form of its Schur
block σ.
Positivity #
The Schur block of a positive definite doubled block matrix is positive semidefinite.
The skew-corrected Schur form of a symmetric positive definite doubled block matrix is positive semidefinite, at every value of the skew parameter.
The conjugated skew-corrected form is symmetric positive semidefinite.
This is what makes specBound of it the spectral norm |·| written in the
intrinsic contrast of the reference block and in e.Theta.m.
The conjugated skew-corrected form is symmetric in the ambient encoding.