aequationes mathematicae

, Volume 40, Issue 1, pp 89–93 | Cite as

Matrix versions of the Cauchy and Kantorovich inequalities

  • A. W. Marshall
  • I. Olkin
Research Papers


A version of Cauchy's inequality is obtained which relates two matrices by an inequality in the sense of the Loewner ordering. In that ordering a symmetric idempotent matrix is dominated by the identity matrix and this fact yields a simple proof.

A consequence of this matrix Cauchy inequality leads to a matrix version of the Kantorovich inequality, again in the sense of Loewner.

AMS (1980) subject classification

Primary 15A45 Secondary 26D15 


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Copyright information

© Birkhäuser Verlag 1990

Authors and Affiliations

  • A. W. Marshall
    • 1
    • 2
    • 3
  • I. Olkin
    • 1
    • 2
    • 3
  1. 1.Department of StatisticsUniversity of British ColumbiaVancouverCanada
  2. 2.Department of StatisticsStanford UniversityStanfordUSA
  3. 3.Department of MathematicsWestern Washington UniversityBellinghamUSA

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