대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제35권3호
  • /
  • Pages.917-925
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  • 2020
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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POSITIVENESS FOR THE RIEMANNIAN GEODESIC BLOCK MATRIX

  • Hwang, Jinmi (Department of Mathematics Chungbuk National University) ;
  • Kim, Sejong (Department of Mathematics Chungbuk National University)
  • 투고 : 2020.02.04
  • 심사 : 2020.05.22
  • 발행 : 2020.07.31
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초록

It has been shown that the geometric mean A#B of positive definite Hermitian matrices A and B is the maximal element X of Hermitian matrices such that $$\(\array{A&X\\X&B}\)$$ is positive semi-definite. As an extension of this result for the 2 × 2 block matrix, we consider in this article the block matrix [[A#wijB]] whose (i, j) block is given by the Riemannian geodesics of positive definite Hermitian matrices A and B, where wij ∈ ℝ for all 1 ≤ i, j ≤ m. Under certain assumption of the Loewner order for A and B, we establish the equivalent condition for the parameter matrix ω = [wij] such that the block matrix [[A#wijB]] is positive semi-definite.

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참고문헌

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