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The problem of reducing polynomial matrices to canonical form by using semiscalar equivalent transformations is studied. This problem is wild as a whole. However, it is tame in some special cases. In the paper, classes of polynomial matrices are singled out for which canonical forms with respect to semiscalar equivalence are indicated. We use this tool to construct a canonical form for the families of coefficients corresponding to the polynomial matrices. This form enables one to solve the classification problem for families of numerical matrices up to similarity. 相似文献
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B. Z. Shavarovskii 《Mathematical Notes》1998,64(5):663-673
The structure of polynomial matrices in connection with their reducibility by semiscalar-equivalent transformations and similarity
transformations to simpler forms is considered. In particular, the canonical form of polynomial matrices without multiple
characteristic roots with respect to the above transformations is indicated. This allows one to establish a canonical form
with respect to similarity for a certain type of finite collections of numerical matrices.
Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 769–782, November, 1998. 相似文献
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B. Z. Shavarovskii 《Ukrainian Mathematical Journal》1999,51(8):1291-1295
By using the transformationsSA(x)R(x), whereS andR(x) are invertible matrices, we reduce a polynomial matrixA(x) whose elementary divisors are pairwise relatively prime to a direct sum of irreducible triangular summands with invariant factors on the principal diagonals. Institute of Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, L’vov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1144–1148, August, 1999. 相似文献
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B. Z. Shavarovskii 《Computational Mathematics and Mathematical Physics》2006,46(8):1283-1292
A certain standard form is found for a complex matrix with respect to equivalent transformations by quasi-diagonal matrices. The solvability of certain matrix equations in the rings of quasi-diagonal matrices is examined using this standard form. 相似文献
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B. Z. Shavarovskii 《Mathematical Notes》2000,68(4):507-518
It is shown that any regular polynomial matrix over the field of complex numbers that has at most one elementary divisor of degree 3 and whose remaining elementary divisors are of degree no greater than 2 can be factored into linear regular factors. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 593–607, October, 2000. 相似文献
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B. Z. Shavarovskii 《Computational Mathematics and Mathematical Physics》2007,47(12):1902-1911
Two classes of matrix polynomial equations with commuting coefficients are examined. It is shown that the equations in one class have complete sets of solutions, whereas the equations in the other class are unsolvable. A method is given for finding the solution set of an equation in the former class. 相似文献
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For a certain class of polynomial matrices A(x), we consider transformations SA(x)R(x) with invertible matrices S and R(x), i.e., the so-called semiscalarly equivalent transformations. We indicate necessary and sufficient conditions for this type of equivalence of matrices. We introduce the notion of quasidiagonal equivalence of numerical matrices. We establish the relationship between the semiscalar and quasidiagonal equivalences and the problem of matrix pairs. 相似文献
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