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On the eigenvalues of matrices with given upper triangular part

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Abstract

We give necessary and sufficient conditions for a set of numbers to be the eigenvalues of a completion of a matrix prescribed in its upper triangular part.

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References

  • [BFT1] H. Bercovici, C. Foias and A. Tannenbaum: A Spectral Commutant Lifting Theorem. Transactions of AMS (to appear).

  • [BFT2] H. Bercovici, C. Foias and A. Tannenbaum: On Spectral Tangential Nevanlinna-Pick Interpolation (preprint).

  • [D1] G. N. De Oliveira: Matrices with Prescribed Characteristic Polynomial and Several Prescribed Submatrices. Lin. and Mult. Alg. 2, 357–364 (1975).

    Google Scholar 

  • [D2] G. N. De Oliveira: Matrices with Prescribed Characteristic Polynomial and Principal Blocks II. Lin. Alg. and Appl. 47, 35–40 (1982).

    Google Scholar 

  • [GKS] I. Gohberg, M.A. Kaashoek and F. van Schagen: Eigenvalues of Completions of Submatrices. Lin. and Mult. Alg. 25, 55–70 (1989).

    Google Scholar 

  • [HJ] R. A. Horn and C. R. Johnson: Matrix Analysis. Cambridge, 1985.

  • [JS] C. R. Johnson and H. M. Shapiro: Mathematical Aspects of the Relative Gain Array, (A oA −T). SIAM J. Alg. Disc. Meth. 7, 627–644 (1986).

    Google Scholar 

  • [M] L. Mirsky: Matrices with Prescribed Characteristic Roots and Diagonal Elements. J. London Math. Soc. 33 (1958), 14–21.

    Google Scholar 

  • [R] H. H. Rosenbrock: State Space and Multivariable Theory. Nelson, London 1970.

    Google Scholar 

  • [S1] F. C. Silva: Matrices with Prescribed Eigenvalues and Principal Submatrices. Lin. Alg. and Appl. 92, 241–250 (1987).

    Google Scholar 

  • [S2] F. C. Silva: Matrices with Prescribed Similarity Class and Prescribed Non-principal Submatrix (preprint).

  • [dS] E. Marques de Sà: Imbedding Conditions for λ-Matrices. Lin. Alg. and Appl. 24, 33–50 (1979).

    Google Scholar 

  • [T] R. C. Thompson: Interlacing Inequalities for Invariant Factors. Lin. Alg. and Appl. 24, 1–32 (1979).

    Google Scholar 

  • [Z1] I. Zaballa: Matrices with Prescribed Rows and Invariant Factors. Lin. Alg. and Appl. 87, 113–146 (1987).

    Google Scholar 

  • [Z2] I. Zaballa: Matrices with Prescribed Invariant Factors and Off-Diagonal Submatrices. Lin. and Mult. Alg. 25, 39–54 (1989).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics Virginia Polytechnic Institute and State University, 24061, Blacksburg, Virginia

    J. A. Ball

  2. School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, 69978, Tel-Aviv, Israel

    I. Gohberg

  3. Department of Mathematics, The College of William and Mary, 23187-8795, Williamsburg, Virginia

    L. Rodman & T. Shalom

Authors
  1. J. A. Ball
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  2. I. Gohberg
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  3. L. Rodman
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  4. T. Shalom
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Additional information

Partially supported by the NSF Grant DMS-8701615-02

Partially supported by the NSF Grant DMS-8802836 and United States-Israel Binational Science Foundation Grant 88-00304/I.

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Ball, J.A., Gohberg, I., Rodman, L. et al. On the eigenvalues of matrices with given upper triangular part. Integr equ oper theory 13, 488–497 (1990). https://doi.org/10.1007/BF01210399

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  • DOI: https://doi.org/10.1007/BF01210399

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