Explicit solution of the inverse eigenvalue problem of real symmetric matrices and its application to electrical network synthesis. (English) Zbl 1155.15007
Summary: A novel procedure for the explicit construction of the entries of real symmetric matrices with assigned spectrum and the entries of the corresponding orthogonal modal matrices is presented. The inverse eigenvalue problem of symmetric matrices with some specific sign patterns (including hyperdominant one) is explicitly solved too. It has been shown to arise thereof a possibility of straightforward solving the inverse eigenvalue problem of symmetric hyperdominant matrices with assigned nonnegative spectrum. The results obtained are applied thereafter in the synthesis of driving-point immittance functions of transformerless, common-ground, two-element-kind RLC networks and in generation of their equivalent realizations.
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A29 | Inverse problems in linear algebra |
| 78A55 | Technical applications of optics and electromagnetic theory |
References:
| [1] | L. Mirsky, “Matrices with prescribed characteristic roots and diagonal elements,” Journal of the London Mathematical Society, vol. 33, pp. 14-21, 1958. · Zbl 0101.25303 · doi:10.1112/jlms/s1-33.1.14 |
| [2] | A. J. Schneider, “Construction of matrices having certain sign-pattern and prescribed eigenvalues by orthogonal transformations,” IEEE Transactions on Circuits Theory, vol. 12, no. 3, pp. 419-421, 1965. |
| [3] | H. Hochstadt, “On some inverse problems in matrix theory,” Archiv der Mathematik, vol. 18, pp. 201-207, 1967. · Zbl 0147.27701 · doi:10.1007/BF01899647 |
| [4] | K. P. Hadeler, “Ein inverses Eigenwertproblem,” Linear Algebra and Its Applications, vol. 1, no. 1, pp. 83-101, 1968. · Zbl 0159.03602 · doi:10.1016/0024-3795(68)90051-7 |
| [5] | K. P. Hadeler, “Multiplikative inverse Eigenwertprobleme,” Linear Algebra and Its Applications, vol. 2, no. 1, pp. 65-86, 1969. · Zbl 0182.05201 · doi:10.1016/0024-3795(69)90008-1 |
| [6] | G. N. de Oliveira, “Matrices with prescribed characteristic polynomial and a prescribed submatrix. III,” Monatshefte für Mathematik, vol. 75, no. 5, pp. 441-446, 1971. · Zbl 0239.15006 · doi:10.1007/BF01297013 |
| [7] | J. A. Dias da Silva, “Matrices with prescribed entries and characteristic polynomial,” Proceedings of the American Mathematical Society, vol. 45, no. 1, pp. 31-37, 1974. · Zbl 0309.15005 · doi:10.2307/2040601 |
| [8] | G. N. de Oliveira, “Matrices with prescribed characteristic polynomial and several prescribed submatrices,” Linear and Multilinear Algebra, vol. 2, no. 4, pp. 357-364, 1975. · Zbl 0302.15014 · doi:10.1080/03081087508817081 |
| [9] | L. J. Gray and D. G. Wilson, “Construction of a Jacobi matrix from spectral data,” Linear Algebra and Its Applications, vol. 14, no. 2, pp. 131-134, 1976. · Zbl 0358.15021 · doi:10.1016/0024-3795(76)90020-3 |
| [10] | S. Friedland, “Inverse eigenvalue problems,” Linear Algebra and Its Applications, vol. 17, no. 1, pp. 15-51, 1977. · Zbl 0358.15007 · doi:10.1016/0024-3795(77)90039-8 |
| [11] | G. N. de Oliveira, “Matrices with prescribed characteristic polynomial and principal blocks. II,” Linear Algebra and Its Applications, vol. 47, pp. 35-40, 1982. · Zbl 0494.15006 · doi:10.1016/0024-3795(82)90224-5 |
| [12] | S. E. Sussman-Fort, “The reconstruction of bordered-diagonal and Jacobi matrices from spectral data,” Journal of the Franklin Institute, vol. 314, no. 5, pp. 271-282, 1982. · Zbl 0492.65026 · doi:10.1016/0016-0032(82)90029-1 |
| [13] | I. Zaballa, “Matrices with prescribed rows and invariant factors,” Linear Algebra and Its Applications, vol. 87, pp. 113-146, 1987. · Zbl 0632.15003 · doi:10.1016/0024-3795(87)90162-5 |
| [14] | F. C. Silva, “Matrices with prescribed eigenvalues and principal submatrices,” Linear Algebra and Its Applications, vol. 92, pp. 241-250, 1987. · Zbl 0651.15007 · doi:10.1016/0024-3795(87)90261-8 |
| [15] | W. B. Gragg and W. J. Harrod, “The numerically stable reconstruction of Jacobi matrices from spectral data,” Numerische Mathematik, vol. 44, no. 3, pp. 317-335, 1984. · Zbl 0556.65027 · doi:10.1007/BF01405565 |
| [16] | D. B. Kandić, B. Parlett, B. D. Reljin, and P. M. Vasić, “Explicit construction of hyperdominant symmetric matrices with assigned spectrum,” Linear Algebra and Its Applications, vol. 258, pp. 41-51, 1997. · Zbl 0878.15004 · doi:10.1016/S0024-3795(96)00161-9 |
| [17] | A. J. Schneider, “RC driving-point impedance realization by linear transformations,” IEEE Transactions on Circuits Theory, vol. 13, no. 3, pp. 265-271, 1966. |
| [18] | S. E. Sussman-Fort, “Inductor-capacitor one-ports and inverse eigenvalue problems,” IEEE Transactions on Circuits and Systems, vol. 28, no. 8, pp. 850-853, 1981. |
| [19] | D. B. Kandić and B. D. Reljin, “Class of non-canonic, driving-point immittance realizations of passive, common-ground, transformerless, two-element-kind RLC networks,” International Journal of Circuit Theory and Applications, vol. 22, no. 3, pp. 163-174, 1994. · Zbl 0809.94025 · doi:10.1002/cta.4490220302 |
| [20] | L. Weinberg, Network Analysis and Synthesis, McGraw-Hill, New York, NY, USA, 1961. |
| [21] | B. D. O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern System Theory Approach, Prentice-Hall, Englewood Cliffs, NJ, USA, 1973. |
| [22] | D. B. Kandić and B. D. Reljin, “The application of polynomial matrix factorization in active network synthesis,” The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 24, no. 4, pp. 1120-1141, 2005. · Zbl 1079.78511 · doi:10.1108/03321640510615490 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
