Hermitian, hermitian R-symmetric, and hermitian R-skew symmetric Procrustes problems

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Abstract

Let R ∈; ℂn×nbe a nontrivial unitary involution; i.e., R = R * = R-1≠ ±I. We say that A ∈ ℂn×nis R-symmetric (R-skew symmetric) if RAR = A (RAR = -A). Let script S sign be one of the following subsets of ℂn×n: (i) hermitian matrices; (ii) hermitian R-symmetric matrices; (iii) hermitian R-skew symmetric matrices. Given Z, W ∈ ℂn×m, we characterize the matrices A in script S sign that minimize ∥AZ - W∥ (Frobenius norm), and, given an arbitrary E ∈ ℂn×n, we find the unique matrix among the minimizers of ∥AZ - W∥ in script S sign that minimizes ∥A - E∥. We also obtain necessary and sufficient conditions for existence of A ∈ script S sign such that AZ = W, and, assuming that the conditions are satisfied, characterize the set of all such A. © 2004 Elsevier Inc. All rights reserved.

Author-supplied keywords

  • Frobenius norm
  • Hermitian
  • Involution
  • Moore-Penrose inverse
  • Procrustes problem
  • R-skew symmetric
  • R-symmetric
  • Singular value decomposition

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Authors

  • William F. Trench

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