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Matrices with prescribed rows and invariant factors. (English) Zbl 0632.15003
This paper contains a most interesting and beautiful result. About nine years ago E. M. de Sá [ibid. 24, 33-50 (1979; Zbl 0395.15009)] and R. C. Thompson [ibid. 24, 1-31 (1979; Zbl 0395.15003)] found the complete relations between the invariant polynomials of a matrix (over a field) and the invariant polynomials of one of its principal submatrices. Their result is known as Sá-Thompson interlacing inequalities and raised some questions so far unsettled. The present author tackles and solves another problem of the same type: consider the matrix (over a field) $G=\left[ \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right]$ where A and D are square blocks, A and B are known and C and D are unknown. Find a necessary and sufficient condition for the existence of C and D so that G has prescribed invariant polynomials.
It cannot be said that the condition presented in this paper is too involved to be stated here. What prevents the reviewer from stating it is the need of explaining the notation. However the reviewer would like to mention that the necessary and sufficient condition consists of some interlacing inequalities similar to those of Sá and Thompson (as expected) together with a majorization relation involving the controllability indices of the pair (A,B) and the degrees of certain polynomials.
Reviewer: G.N.de Oliveira

MSC:
 15A15 Determinants, permanents, traces, other special matrix functions 15A45 Miscellaneous inequalities involving matrices 15A21 Canonical forms, reductions, classification
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References:
 [1] Barnett, S., Polynomials and linear control system, (1983), Marcel Dekker New York [2] Brunovsky, P., A classification of linear controllable systems, Kybernetika (Prague), 3, 6, 173-188, (1970) · Zbl 0199.48202 [3] Coppel, W.A., Linear systems: some algebraic aspects, Linear algebra appl., 40, 257-273, (1981) [4] Djaferis, T.E.; Mitter, S.K., Some generic invariant factor assignment results using dynamic output feedback, Linear algebra appl., 50, 103-131, (1983) · Zbl 0514.93032 [5] Friedland, S., Spectral theory of matrices I. general matrices, (1980), Mathematics Research Center, Univ. of Wisconsin Madison [6] Gantmacher, F.R., Théorie des matrices, (1966), Dunod Paris, Tome I · Zbl 0136.00410 [7] Gantmacher, F.R., Théorie des matrices, (1966), Dunod Paris, Tome II · Zbl 0136.00410 [8] Hardy, G.H.; Littlewood, J.E.; Polya, G., Inequalities, (1967), Cambridge U.P · Zbl 0634.26008 [9] Lancaster, P., Lambda-matrices and vibrating systems, (1966), Pergamon Oxford · Zbl 0146.32003 [10] Lancaster, P., Theory of matrices, (1969), Academic London · Zbl 0186.05301 [11] Markus, A.S.; Parilis, E.E., The change of the Jordan structure of a matrix under small perturbations, Linear algebra appl., 54, 139-152, (1983) · Zbl 0524.15010 [12] Marshall, A.W.; Olkin, I., Inequalities: theory of majorization and its applications, (1979), Academic London · Zbl 0437.26007 [13] de Oliveira, G.N., Matrices with prescribed characteristic polynomial and a prescribed submatrix III, Monasth. math., 75, 441-446, (1971) · Zbl 0239.15006 [14] Rosenbrock, H.H., State-space and multivariable theory, (1970), Thomas Nelson London · Zbl 0246.93010 [15] Popov, V.M., Invariant description of linear time-invariant controllable systems, SIAM J. control, 15, 2, 252-264, (1972) · Zbl 0251.93013 [16] Marques de Sà, E., Imbedding conditions for λ-matrices, Linear algebra appl., 24, 33-50, (1979) · Zbl 0395.15009 [17] Marques de Sà, E., Imersão de matrizes e entrelaçamento de factores invariantes, () [18] Thompson, R.C., Interlacing inequalities for invariant factors, Linear algebra appl., 24, 1-32, (1979) · Zbl 0395.15003 [19] Wimmer, H.K., Existenzsätze in der theorie der matrizen und lineare kontroll-theorie, Monasth. math., 78, 256-263, (1974) · Zbl 0288.15014 [20] Wolovich, A., Linear multivariable systems, (1974), Springer New York · Zbl 0291.93002
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