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Solution of the Problem of Optimal Diagonal Scaling for Quasireal Hermitian Positive-Definite 3×3 Matrices

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Abstract

The paper completely solves the problem of optimal diagonal scaling for quasireal Hermitian positive-definite matrices of order 3. In particular, in the most interesting irreducible case, it is demonstrated that for any matrix C from the class considered there is a uniquely determined optimally scaled matrix D *0 CD0 of one of the four canonical types. Formulas for the entries of the diagonal matrix D0 are presented, as well as formulas for the eigenvalues and eigenvectors of D *0 CD0 and for the optimal condition number of C, which is equal to k(D *0 CD0). The optimality of the Jacobi scaling is analyzed. Bibliography: 10 titles.

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REFERENCES

  1. F. L. Bauer, “Optimally scaled matrices,” Numer. Math., 5, 73–87 (1963).

    Article  MATH  MathSciNet  Google Scholar 

  2. G. E. Forsythe and E. G. Straus, “On best conditioned matrices,” Proc. Amer. Math. Soc., 6, 340–345 (1955).

    MathSciNet  Google Scholar 

  3. R. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press (1991).

  4. L. Yu. Kolotilina, “Optimally conditioned block 2 × 2 matrices,” Zap. Nauchn. Semin. POMI, 268, 72–85 (2000).

    MATH  Google Scholar 

  5. L. Yu. Kolotilina, “On the extreme eigenvalues of block 2 × 2 Hermitian matrices,” Zap. Nauchn. Semin. POMI, 296, 27–38 (2003).

    Google Scholar 

  6. L. Yu. Kolotilina, “Bounds for the extreme eigenvalues of block 2 × 2 Hermitian matrices,” Zap. Nauchn. Semin. POMI, 301, 172–194 (2003).

    Google Scholar 

  7. C. McCarthy and G. Strang, “Optimal conditioning of matrices,” SIAM J. Numer. Anal., 10, 370–388 (1973).

    Article  MathSciNet  Google Scholar 

  8. A. Shapiro, “Optimally scaled matrices, necessary and sufficient conditions,” Numer. Math., 39, 239–245 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Shapiro, “Optimal block diagonal l 2-scaling of matrices,” SIAM J. Numer. Math., 22, 81–94 (1985).

    MATH  Google Scholar 

  10. A. Van Der Sluis, “Condition numbers and equilibration of matrices,” Numer. Math., 14, 14–23 (1969).

    Article  MATH  MathSciNet  Google Scholar 

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  1. St. Petersburg Department, Steklov Mathematical Institute, St. Petersburg, Russia

    L. Yu. Kolotilina

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  1. L. Yu. Kolotilina
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 309, 2004, pp. 84–126.

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Kolotilina, L.Y. Solution of the Problem of Optimal Diagonal Scaling for Quasireal Hermitian Positive-Definite 3×3 Matrices. J Math Sci 132, 190–213 (2006). https://doi.org/10.1007/s10958-005-0488-1

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  • DOI: https://doi.org/10.1007/s10958-005-0488-1

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