A new proof of the Benedetti's inequality and some applications to perturbation to real eigenvalues and singular values

• Humberto Sarria ^[1] ; Juan Carlos Martínez ^[2]
  1. [1] Universidad Nacional de Colombia

     Universidad Nacional de Colombia

     Colombia

     
  2. [2] Universidad del Rosario

     Universidad del Rosario

     Colombia

     
• Localización: Boletín de matemáticas, ISSN 0120-0380, ISSN-e 2357-6529, Vol. 23, Nº. 2, 2016, págs. 105-114
• Idioma: inglés
• Títulos paralelos:
  • Una nueva prueba de desigualdad de Benedetti y algunas aplicaciones a la perturbación de valores propios reales y valores singulares
• Enlaces
  • Texto completo (pdf)
• Resumen

  • Using the standard deviation of the real samples μn ≥ … ≥ μ1 and λn ≥ … ≥ λ1, we refine the Chebyshev's inequality (refer to [5]), As a consequence, we obtain a new proof of the Benedetti's inequality (refer to [1], [2] and [4]) where Cov[μ, λ], s(μ) and s(λ) denote the covariance, and the standard deviations (≠ 0) of the sample vectors μ = (μ1, …, μn) and λ = (λ1, …, λn), respectively.

    We can also get very interesting applications to eigenvalues and singular values perturbation theory. For some kinds of matrices, the result that we present improves the well known Homand-Weiland's inequality.