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Notes: The Fuping Complex and the adjoining Wutai and Hengshan Complexes are located in the central zone of the North China craton. The dominant rock types in the Fuping Complex are high-grade tonalitic–trondhjemitic–granodioritic (TTG) gneisses, with minor amounts of mafic granulites, syntectonic granitic rocks and supracrustal rocks. The petrological evidence from the mafic granulites indicates three stages of metamorphic evolution. The M1 stage is represented by garnet porphyroblasts and matrix plagioclase, quartz, orthopyroxene, clinopyroxene and hornblende. Orthopyroxene+plagioclase symplectites and clinopyroxene+plagioclase±orthopyroxene coronas formed in response to decompression during M2 following the peak metamorphism at M1. Hornblende+plagioclase symplectites formed as a result of further isobaric cooling and retrograde metamorphism during M3. The P–T  estimates using TWQ thermobarometry are: 900–950 °C and 8.0–8.5 kbar for the peak assemblage (M1), based on the core compositions of garnet, matrix pyroxene and plagioclase; 700–800 °C and 6.0–7.0 kbar for the pyroxene+plagioclase symplectites or coronas (M2); and 550–650 °C and 5.3–6.3 kbar for the hornblende+plagioclase symplectites (M3), based on garnet rim and corresponding symplectic mineral compositions. These P–T  estimates define a clockwise P–T  path involving near-isothermal decompression for the Fuping Complex, similar to the P–T  path estimated for the metapelitic gneisses. The inferred P–T  path suggests that the Fuping Complex underwent initial crustal thickening, subsequent exhumation, and finally cooling and retrogression. This tectonothermal path is similar to P–T  paths inferred for the Wutai and Hengshan Complexes and other tectonic units in the central zone of the North China craton, but different from anti-clockwise P–T paths estimated for the basement rocks in the eastern and western zones of the craton. Based on lithological, structural, metamorphic and geochronological data, the eastern and western zones of the craton are considered to represent two different Archean to Paleoproterozoic continental blocks that amalgamated along the central zone at the end of Paleoproterozoic. The P–T paths of the Fuping Complex and other tectonic units in the central zone record the collision between the eastern and western zones that led to the final assembly of the North China craton at c. 1800 Ma.

Notes: Abstract For two Hermitian matrices A and B, at least one of which is positive semidefinite, we give upper and lower bounds for each eigenvalue of AB in terms of the eigenvalues of A and B. For two complex matrices A,B with known singular values, upper and lower bounds are deduced for each singular value of AB.