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Georg Heinig November 24, 1947 - May 10, 2005 A Personal Memoir and Appreciation

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Numerical Methods for Structured Matrices and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 199))

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Abstract

On May 10, 2005, Georg Heinig, an excellent mathematician died unexpectedly at the age of 57. He was a world leader in the field of structured matrices. As associate editor of the journal Linear Algebra and its Applications since his appointment in 1991 he contributed much to the journal’s success by his valuable and extensive work. In what follows I want to try to capture some aspects of this mathematical life and his personality.

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References

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List of Georg Heinig’s (refereed) publications (chronologically ordered, including one monograph [85, 88], two edited proceedings [14, 33], and one book translation [30])

  1. G. Heinig, K. Rost, Split algorithms for centrosymmetric Toeplitz-plus-Hankel matrices with arbitrary rank profile, 129–146, Oper. Theory Adv. Appl., 171, Birkhäuser, Basel, 2007.

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  2. G. Heinig, K. Rost, Schur-type algorithms for the solution of Hermitian Toeplitz systems via factorization, 233–252, Oper. Theory Adv. Appl., 160, Birkhäuser, Basel, 2005.

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  3. G. Codevico, G. Heinig, M. Van Barel, A superfast solver for real symmetric Toeplitz systems using real trigonometric transformations. Numer. Linear Algebra Appl. 12 (2005), 699–713.

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  4. G. Heinig, K. Rost, Fast “split” algorithms for Toeplitz and Toeplitz-plus-Hankel matrices with arbitrary rank profile. Proceedings of the International Conference on Mathematics and its Applications (ICMA 2004), 285–312, Kuwait, 2005.

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  8. G. Heinig, K. Rost, Split algorithms for skewsymmetric Toeplitz matrices with arbitrary rank profile. Theoret. Comput. Sci. 315 (2004), no. 2-3, 453–468.

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  9. G. Heinig, K. Rost, New fast algorithms for Toeplitz-plus-Hankel matrices. SIAM J. Matrix Anal. Appl. 25 (2003), no. 3, 842–857.

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  10. G. Heinig, K. Rost, Fast algorithms for centrosymmetric and centro-skewsymmetric Toeplitz-plus-Hankel matrices. International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001), Numer. Algorithms 33 (2003), no. 1-4, 305–317.

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  11. G. Heinig, Inversion of Toeplitz-plus-Hankel matrices with arbitrary rank profile. Fast algorithms for structured matrices: theory and applications (South Hadley, MA, 2001), 75-89, Contemp. Math., 323, Amer. Math. Soc., Providence, RI, 2003.

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  12. M. Van Barel, G. Heinig, P. Kravanja, A superfast method for solving Toeplitz linear least squares problems. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000), Linear Algebra Appl. 366 (2003), 441–457.

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  13. G. Heinig, K. Rost, Centrosymmetric and centro-skewsymmetric Toeplitz-plus-Hankel matrices and Bezoutians. Special issue on structured matrices: analysis, algorithms and applications (Cortona, 2000), Linear Algebra Appl. 366 (2003), 257–281.

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  14. [14] Special issue on structured matrices: analysis, algorithms and applications. Papers from the workshop held in Cortona, September 21-28, 2000. eds. D. Bini, G. Heinig, E. Tyrtyshnikov. Linear Algebra Appl. 366 (2003). Elsevier Science B.V., Amsterdam, 2003.

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  15. G. Heinig, K. Rost, Fast algorithms for skewsymmetric Toeplitz matrices. Toeplitz matrices and singular integral equations (Pobershau, 2001), 193-208, Oper. Theory Adv. Appl., 135, Birkhäuser, Basel, 2002.

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  16. G. Heinig, On the reconstruction of Toeplitz matrix inverses from columns. Linear Algebra Appl. 350 (2002), 199–212.

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  17. G. Heinig, K. Rost, Centro-symmetric and centro-skewsymmetric Toeplitz matrices and Bezoutians. Special issue on structured and infinite systems of linear equations. Linear Algebra Appl. 343/344 (2002), 195–209.

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  18. G. Heinig, Kernel structure of Toeplitz-plus-Hankel matrices. Linear Algebra Appl. 340 (2002), 1–13.

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  19. G. Heinig, Fast and superfast algorithms for Hankel-like matrices related to orthogonal polynomials. Numerical analysis and its applications (Rousse, 2000), 385-392, Lecture Notes in Comput. Sci., 1988, Springer, Berlin, 2001.

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  20. M. VanBarel, G. Heinig, P. Kravanja, An algorithm based on orthogonal polynomial vectors for Toeplitz least squares problems. Numerical analysis and its applications (Rousse, 2000), 27-34, Lecture Notes in Comput. Sci., 1988, Springer, Berlin, 2001.

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  21. M. VanBarel, G. Heinig, P. Kravanja, A stabilized superfast solver for nonsymmetric Toeplitz systems. SIAM J. Matrix Anal. Appl. 23 (2001), no. 2, 494–510.

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  22. G. Heinig, K. Rost, Efficient inversion formulas for Toeplitz-plus-Hankel matrices using trigonometric transformations. Structured matrices in mathematics, computer science, and engineering, II (Boulder, CO, 1999), 247-264, Contemp. Math., 281, Amer. Math. Soc., Providence, RI, 2001.

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  23. G. Heinig, Stability of Toeplitz matrix inversion formulas. Structured matrices in mathematics, computer science, and engineering, II (Boulder, CO, 1999), 101-116, Contemp. Math., 281, Amer. Math. Soc., Providence, RI, 2001.

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  24. G. Heinig, V. Olshevsky, The Schur algorithm for matrices with Hessenberg displacement structure. Structured matrices in mathematics, computer science, and engineering, II (Boulder, CO, 1999), 3-15, Contemp. Math., 281, Amer. Math. Soc., Providence, RI, 2001.

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  25. G. Heinig, Not every matrix is similar to a Toeplitz matrix. Proceedings of the Eighth Conference of the International Linear Algebra Society (Barcelona, 1999). Linear Algebra Appl. 332/334 (2001), 519–531.

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  26. G. Heinig, Chebyshev-Hankel matrices and the splitting approach for centrosymmetric Toeplitz-plus-Hankel matrices. Linear Algebra Appl. 327 (2001), no. 1-3, 181–196.

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  27. S. Feldmann, G. Heinig, Partial realization for singular systems in standard form. Linear Algebra Appl. 318 (2000), no. 1-3, 127–144.

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  28. G. Heinig, K. Rost, Representations of inverses of real Toeplitz-plus-Hankel matrices using trigonometric transformations. Large-scale scientific computations of engineering and environmental problems, II (Sozopol, 1999), 80-86, Notes Numer. Fluid Mech., 73, Vieweg, Braunschweig, 2000.

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  29. M. Van Barel, G. Heinig, P. Kravanja, Least squares solution of Toeplitz systems based on orthogonal polynomial vectors. Advanced Signal Processing Algorithms, Architectures, and Implementations X. ed. F.T. Luk, Vol 4116, Proceedings of SPIE (2000), 167–172.

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  30. V. Maz’ya, S. Nazarov, B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I. Translated from the German by Georg Heinig and Christian Posthoff. Operator Theory: Advances and Applications, 111. Birkhäuser, Basel, 2000.

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  31. G. Heinig, K. Rost, Hartley transform representations of symmetric Toeplitz matrix inverses with application to fast matrix-vector multiplication. SIAMJ. Matrix Anal. Appl. 22 (2000), no. 1, 86–105.

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  32. G. Heinig, K. Rost, Hartley transform representations of inverses of real Toeplitzplus-Hankel matrices. Proceedings of the International Conference on Fourier Analysis and Applications (Kuwait, 1998). Numer. Funct. Anal. Optim. 21 (2000), no. 1-2, 175–189.

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  33. Proceedings of the International Conference on Fourier Analysis and Applications. Held at Kuwait University, Kuwait, May 3-6, 1998, Eds. F.Al-Musallam, A.Böttcher, P.Butzer, G.Heinig, Vu Kim Tuan. Numer. Funct. Anal. Optim. 21 (2000), no. 1-2. Marcel Dekker, Inc., Monticello, NY, 2000.

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  34. G. Heinig, F. Al-Musallam, Hermite’s formula for vector polynomial interpolation with applications to structured matrices. Appl. Anal. 70 (1999), no. 3-4, 331–345.

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  35. S. Feldmann, G. Heinig, Parametrization of minimal rank block Hankel matrix extensions and minimal partial realizations. Integral Equations Operator Theory 33 (1999), no. 2, 153–171.

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  36. G. Heinig, K. Rost, DFT representations of Toeplitz-plus-Hankel Bezoutians with application to fast matrix-vector multiplication. ILAS Symposium on Fast Algorithms for Control, Signals and Image Processing (Winnipeg, MB, 1997). Linear Algebra Appl. 284 (1998), no. 1-3, 157–175.

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  37. G. Heinig, Matrices with higher-order displacement structure. Linear Algebra Appl. 278 (1998), no. 1-3, 295–301.

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  38. G. Heinig, A. Bojanczyk, Transformation techniques for Toeplitz and Toeplitz-plus-Hankel matrices. II. Algorithms. Linear Algebra Appl. 278 (1998), no. 1-3, 11–36.

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  39. G. Heinig, Properties of “derived” Hankel matrices. Recent progress in operator theory (Regensburg, 1995), 155-170, Oper. Theory Adv. Appl., 103, Birkhäuser, Basel, 1998.

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  40. G. Heinig, K. Rost, Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication. Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996). Linear Algebra Appl. 275/276 (1998), 225–248.

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  41. G. Heinig, F.Al-Musallam, Lagrange’s formula for tangential interpolation with application to structured matrices. Integral Equations Operator Theory 30 (1998), no. 1, 83–100.

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  42. G. Heinig, Generalized Cauchy-Vandermonde matrices. Linear Algebra Appl. 270 (1998), 45–77.

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  43. G. Heinig, The group inverse of the transformation S(X) = 3DAX − XB. Linear Algebra Appl. 257 (1997), 321–342.

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  44. G. Heinig, A. Bojanczyk, Transformation techniques for Toeplitz and Toeplitz-plus-Hankel matrices. I. Transformations. Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995). Linear Algebra Appl. 254 (1997), 193–226.

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  45. G. Heinig, L.A. Sakhnovich, I.F. Tidniuk, Paired Cauchy matrices. Linear Algebra Appl. 251 (1997), 189–214.

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  46. G. Heinig, Solving Toeplitz systems after extension and transformation. Toeplitz matrices: structures, algorithms and applications (Cortona, 1996). Calcolo 33 (1998), no. 1-2, 115–129.

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  47. S. Feldmann, G. Heinig, On the partial realization problem for singular systems. Proceedings of the 27th Annual Iranian Mathematics Conference (Shiraz, 1996), 79–100, Shiraz Univ., Shiraz, 1996.

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  48. S. Feldmann, G. Heinig, Vandermonde factorization and canonical representations of block Hankel matrices. Proceedings of the Fourth Conference of the International Linear Algebra Society (ILAS) (Rotterdam, 1994). Linear Algebra Appl. 241/243 (1996), 247–278.

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  49. G. Heinig, Inversion of generalized Cauchy matrices and other classes of structured matrices. Linear algebra for signal processing (Minneapolis, MN, 1992), 63–81, IMA Vol. Math. Appl., 69, Springer, New York, 1995.

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  50. G. Heinig, Matrix representations of Bezoutians. Special issue honoring Miroslav Fiedler and Vlastimil Pt’ak. Linear Algebra Appl. 223/224 (1995), 337–354.

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  51. G. Heinig, K. Rost, Recursive solution of Cauchy-Vandermonde systems of equations. Linear Algebra Appl. 218 (1995), 59–72.

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  52. G. Heinig, Generalized inverses of Hankel and Toeplitz mosaic matrices. Linear Algebra Appl. 216 (1995), 43–59.

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  53. G. Heinig, F. Hellinger, Displacement structure of generalized inverse matrices. Generalized inverses (1993). Linear Algebra Appl. 211 (1994), 67–83.

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  54. G. Heinig, F. Hellinger, The finite section method for Moore-Penrose inversion of Toeplitz operators. Integral Equations Operator Theory 19 (1994), no. 4, 419–446.

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  55. G. Heinig, F. Hellinger, Displacement structure of pseudoinverses. Second Conference of the International Linear Algebra Society (ILAS) (Lisbon, 1992). Linear Algebra Appl. 197/198 (1994), 623–649.

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  56. S. Feldmann, G. Heinig, Uniqueness properties of minimal partial realizations. Linear Algebra Appl. 203/204 (1994), 401–427.

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  57. G. Heinig, F. Hellinger, Moore-Penrose inversion of square Toeplitz matrices. SIAM J. Matrix Anal. Appl. 15 (1994), no. 2, 418–450.

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  58. A.W. Boja’nczyk, G. Heinig, A multi-step algorithm for Hankel matrices. J. Complexity 10 (1994), no. 1, 142–164.

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  59. G. Heinig, F. Hellinger, On the Bezoutian structure of the Moore-Penrose inverses of Hankel matrices. SIAM J. Matrix Anal. Appl. 14 (1993), no. 3, 629–645.

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  60. T. Finck, G. Heinig, K. Rost, An inversion formula and fast algorithms for Cauchy-Vandermonde matrices. Linear Algebra Appl. 183 (1993), 179–191.

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  61. G. Heinig, P. Jankowski, Kernel structure of block Hankel and Toeplitz matrices and partial realization. Linear Algebra Appl. 175 (1992), 1–30.

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  62. G. Heinig, Inverse problems for Hankel and Toeplitz matrices. Linear Algebra Appl. 165 (1992), 1–23.

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  63. G. Heinig, Fast algorithms for structured matrices and interpolation problems. Algebraic computing in control (Paris, 1991), 200-211, Lecture Notes in Control and Inform. Sci., 165, Springer, Berlin, 1991.

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  64. G. Heinig, On structured matrices, generalized Bezoutians and generalized Christoffel-Darboux formulas. Topics in matrix and operator theory (Rotterdam, 1989), 267-281, Oper. Theory Adv. Appl., 50, Birkhäuser, Basel, 1991.

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  65. G. Heinig, Formulas and algorithms for block Hankel matrix inversion and partial realization. Signal processing, scattering and operator theory, and numerical methods (Amsterdam, 1989), 79–90, Progr. Systems Control Theory, 5, Birkhäuser Boston, Boston, MA, 1990.

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  66. G. Heinig, P. Jankowski, Parallel and superfast algorithms for Hankel systems of equations. Numer. Math. 58 (1990), no. 1, 109–127.

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  67. G. Heinig, P. Jankowski, Fast algorithms for the solution of general Toeplitz systems. Wiss. Z. Tech. Univ. Karl-Marx-Stadt 32 (1990), no. 1, 12–17.

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  68. G. Heinig, K. Rost, Matrices with displacement structure, generalized Bezoutians, and Moebius transformations. The Gohberg anniversary collection, Vol. I (Calgary, AB, 1988), 203-230, Oper. Theory Adv. Appl., 40, Birkhäuser, Basel, 1989.

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  69. G. Heinig, K. Rost, Inversion of matrices with displacement structure. Integral Equations Operator Theory 12 (1989), no. 6, 813–834.

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  70. G. Heinig, W. Hoppe, K. Rost, Structured matrices in interpolation and approximation problems. Wiss. Z. Tech. Univ. Karl-Marx-Stadt 31 (1989), no. 2, 196–202.

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  71. G. Heinig, K. Rost, Matrix representations of Toeplitz-plus-Hankel matrix inverses. Linear Algebra Appl. 113 (1989), 65–78.

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  72. G. Heinig, T. Amdeberhan, On the inverses of Hankel and Toeplitz mosaic matrices. Seminar Analysis (Berlin, 1987/1988), 53–65, Akademie-Verlag, Berlin, 1988.

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  73. G. Heinig, P. Jankowski, K. Rost, Tikhonov regularisation for block Toeplitz matrices. Wiss. Z. Tech. Univ. Karl-Marx-Stadt 30 (1988), no. 1, 41–45.

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  74. G. Heinig, K. Rost, On the inverses of Toeplitz-plus-Hankel matrices. Linear Algebra Appl. 106 (1988), 39–52.

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  75. G. Heinig, P. Jankowski, K. Rost, Fast inversion algorithms of Toeplitz-plus-Hankel matrices. Numer. Math. 52 (1988), no. 6, 665–682.

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  76. G. Heinig, Structure theory and fast inversion of Hankel striped matrices. I. Integral Equations Operator Theory 11 (1988), no. 2, 205–229.

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  77. G. Heinig, K. Rost, Inversion of generalized Toeplitz-plus-Hankel matrices. Wiss. Z. Tech. Univ. Karl-Marx-Stadt 29 (1987), no. 2, 209–211.

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  78. G. Heinig, U. Jungnickel, Hankel matrices generated by Markov parameters, Hankel matrix extension, partial realization, and Pad’e-approximation. Operator theory and systems (Amsterdam, 1985), 231–253, Oper. Theory Adv. Appl., 19, Birkhäuser, Basel, 1986.

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  79. G. Heinig, U. Jungnickel, Lyapunov equations for companion matrices. Linear Algebra Appl. 76 (1986), 137–147.

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  80. G. Heinig, U. Jungnickel, Hankel matrices generated by the Markov parameters of rational functions. Linear Algebra Appl. 76 (1986), 121–135.

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  81. G. Heinig, Partial indices for Toeplitz-like operators. Integral Equations Operator Theory 8 (1985), no. 6, 805–824.

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  82. G. Heinig, K. Rost, Fast inversion of Toeplitz-plus-Hankel matrices. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 27 (1985), no. 1, 66–71.

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  83. G. Heinig, U. Jungnickel, On the Bezoutian and root localization for polynomials. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 27 (1985), no. 1, 62–65.

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  84. G. Heinig, B. Silbermann, Factorization of matrix functions in algebras of bounded functions. Spectral theory of linear operators and related topics (Timisoara/Herculane, 1983), 157–177, Oper. Theory Adv. Appl., 14, Birkhäuser, Basel, 1984.

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  85. G. Heinig, K. Rost, Algebraic methods for Toeplitz-like matrices and operators. Oper. Theory Adv. Appl., 13, Birkhäuser, Basel, 1984.

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  86. G. Heinig, U. Jungnickel, On the Routh-Hurwitz and Schur-Cohn problems for matrix polynomials and generalized Bezoutians. Math. Nachr. 116 (1984), 185–196.

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  87. G. Heinig, K. Rost, Schnelle Invertierungsalgorithmen für einige Klassen von Matrizen. (German) [Fast inversion algorithms for some classes of matrices] Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 26 (1984), no. 2, 235–241.

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  88. G. Heinig, K. Rost, Algebraic methods for Toeplitz-like matrices and operators. Mathematical Research, 19, Akademie-Verlag, Berlin, 1984.

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  89. G. Heinig, K. Rost, Invertierung von Toeplitzmatrizen und ihren Verallgemeinerungen. I. Die Methode der UV-Reduktion. (German) [Inversion of Toeplitz matrices and their generalizations. I. The method of UV-reduction] Beiträge Numer. Math. No. 12 (1984), 55–73.

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  90. G. Heinig, Generalized resultant operators and classification of linear operator pencils up to strong equivalence. Functions, series, operators, Vol. I, II (Budapest, 1980), 611–620, Colloq. Math. Soc. J’anos Bolyai, 35, North-Holland, Amsterdam, 1983.

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  91. G. Heinig, Inversion of Toeplitz and Hankel matrices with singular sections. Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 25 (1983), no. 3, 326–333.

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  92. G. Heinig, U. Jungnickel, Zur Lösung von Matrixgleichungen der Form AX−XB = 3DC. (German) [On the solution of matrix equations of the form AX−XB = 3DC] Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 23 (1981), no. 4, 387–393.

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  93. G. Heinig, Linearisierung und Realisierung holomorpher Operatorfunktionen. (German) Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 22 (1980), no. 5, 453–459.

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  94. G. Heinig, K. Rost, Invertierung einiger Klassen von Matrizen und Operatoren. I. Endliche Toeplitzmatrizen und ihre Verallgemeinerungen. (German) [Inversion of some classes of matrices and operators. I. Finite Toeplitz matrices and their generalizations] Wissenschaftliche Informationen [Scientific Information], 12, Technische Hochschule Karl-Marx-Stadt, Sektion Mathematik, Karl-Marx-Stadt, 1979.

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  95. G. Heinig, Bezoutiante, Resultante und Spektralverteilungsprobleme für Operatorpolynome. (German) Math. Nachr. 91 (1979), 23–43.

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  96. G. Heinig, Transformationen von Toeplitz-und Hankelmatrizen. (German) Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 21 (1979), no. 7, 859–864.

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  97. G. Heinig, Invertibility of singular integral operators. (Russian) Soobshch. Akad. Nauk Gruzin. SSR 96 (1979), no. 1, 29–32.

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  98. G. Heinig, Ü ber ein kontinuierliches Analogon der Begleitmatrix eines Polynoms und die Linearisierung einiger Klassen holomorpher Operatorfunktionen. (German) Beiträge Anal. 13 (1979), 111–126.

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  99. G. Heinig, Verallgemeinerte Resultantenbegriffe bei beliebigen Matrixbüscheln. II. Gemischter Resultantenoperator. (German) Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 20 (1978), no. 6, 701–703.

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  100. G. Heinig, Verallgemeinerte Resultantenbegriffe bei beliebigen Matrixbüscheln. I. Einseitiger Resultantenoperator. (German) Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 20 (1978), no. 6, 693–700.

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  101. G. Heinig, Endliche Toeplitzmatrizen und zweidimensionale Wiener-Hopf-Operatoren mit homogenem Symbol. II. Über die normale Auflösbarkeit einer Klasse zweidimensionaler Wiener-Hopf Operatoren. (German) Math. Nachr. 82 (1978), 53–68.

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  102. G. Heinig, Endliche Toeplitzmatrizen und zweidimensionale Wiener-Hopf-Operatoren mit homogenem Symbol. I. Eigenschaften endlicher Toeplitzmatrizen. (German) Math. Nachr. 82 (1978), 29–52.

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  103. G. Heinig, K. Rost, Über homogene Gleichungen vom Faltungstyp auf einem endlichen Intervall. (German) Demonstratio Math. 10 (1977), no. 3-4, 791–806.

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  1. Department of Mathematics, Chemnitz University of Technology, D-09107, Chemnitz, Germany

    Karla Rost

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy

    Dario Andrea Bini

  2. Institut für Mathematik, Technische Universität Berlin, Straβe des 17. Juni 136, 10623, Berlin, Germany

    Volker Mehrmann

  3. Department of Mathematics, University of Connecticut, 196 Auditorium Road, U-9, Storrs, CT, 06269, USA

    Vadim Olshevsky

  4. Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina Street, 8, Moscow, 119991, Russia

    Eugene E. Tyrtyshnikov

  5. Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001, Leuven (Heverlee), Belgium

    Marc van Barel

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Rost, K. (2010). Georg Heinig November 24, 1947 - May 10, 2005 A Personal Memoir and Appreciation. In: Bini, D.A., Mehrmann, V., Olshevsky, V., Tyrtyshnikov, E.E., van Barel, M. (eds) Numerical Methods for Structured Matrices and Applications. Operator Theory: Advances and Applications, vol 199. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8996-3_2

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