Modulus of continuity of the matrix absolute value

Rajendra Bhatia ()
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Rajendra Bhatia: Indian Statistical Institute

Indian Journal of Pure and Applied Mathematics, 2010, vol. 41, issue 1, 99-111

Abstract: Abstract Lipschitz continuity of the matrix absolute value |A| = (A*A)1/2 is studied. Let A and B be invertible, and let M 1 = max(‖A‖, ‖B‖), M 2 = max(‖A −1‖, ‖B −1‖). Then it is shown that $$ \left\| { \left| A \right| - \left| B \right| } \right\| \leqslant \left( {1 + log M_1 M_2 } \right) \left\| {A - B} \right\| $$ . A proof is given for the well-known theorem that there is a constant c(n) such that for any two n × n matrices A and B ‖ |A| − |B|‖ ≤ c(n) ‖A − B‖ and the best constant in this inequality is O(log n).

Keywords: Matrix absolute value; perturbation bound; commutator; triangular truncation; Schur product (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s13226-010-0014-0

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