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Upper triangular similarity of upper triangular matrices. (English) Zbl 0881.15011

The author considers the following equivalence relation in the set of all complex upper triangular \(n\times n\) matrices: \(A\) and \(B\) are \(U\)-similar if there exists an invertible upper triangular matrix \(S\) such that \(A=S^{-1} BS\). If \(A\) and \(B\) are \(U\)-similar, then they must have the same diagonal and the same Jordan form. It is known that for \(n\geq 6\) there are infinitely many pairwise non-\(U\)-similar nilpotent upper triangular matrices with the same Jordan form. The author generalizes the concept of Jordan block (called irreducible) and proves that an upper triangular matrix is \(U\)-similar to a “generalized” direct sum of irreducible blocks, where the location and order of the blocks is fixed and each block is uniquely determined up to \(U\)-similarity.

MSC:

15A21 Canonical forms, reductions, classification
15B57 Hermitian, skew-Hermitian, and related matrices
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[1] Bart, H.; Hoogland, H., Complementary triangular forms of pairs of matrices, realizations with prescribed main matrices, and complete factorization of rational matrix functions, Linear Algebra Appl., 103, 193-228 (1988) · Zbl 0645.15012
[2] Bart, H.; Thijsse, G. Ph. A., Complementary triangular forms of upper triangular Toeplitz matrices, Oper. Theory Adv. Appl., 40, 133-149 (1989) · Zbl 0674.15017
[3] Bart, H.; Thijsse, G. Ph. A., Similarity invariants for pairs of upper triangular Toeplitz matrices, Linear Algebra Appl., 147, 17-44 (1991) · Zbl 0715.15005
[4] Bart, H.; Thijsse, G. Ph. A., Eigenspace and Jordan-Chain Techniques for the Description of Complementary Triangular Forms, (Rep. 9353/B (1993), Econometric Inst., Erasmus Univ: Econometric Inst., Erasmus Univ Rotterdam) · Zbl 0685.15010
[5] Djoković, D.ẑ.; Malzan, J., Orbits of nilpotent matrices, Linear Algebra Appl., 32, 157-158 (1980) · Zbl 0436.15015
[6] Higman, G., Enumerating \(p- groups\) I: Inequalities, (Proc. London Math. Soc., 10 (1960)), 24-30, (3) · Zbl 0093.02603
[7] Higman, G., Enumerating \(p- groups\) II: Problems whose solution is PORC, (Proc. London Math. Soc., 10 (1960)), 566-582, (3) · Zbl 0201.36502
[8] Roitman, M., A problem on conjugacy of matrices, Linear Algebra Appl., 19, 87-89 (1978) · Zbl 0371.15008
[9] Thijsse, Ph., Upper Triangular Similarity of Upper Triangular Matrices, (Rep. 9092/A (1990), Econometric Inst., Erasmus Univ: Econometric Inst., Erasmus Univ Rotterdam) · Zbl 0881.15011
[10] Thijsse, Ph., Spectral criteria for complementary triangular forms, Integral Equations Operator Theory, 27, 228-251 (1997)
[11] Vera-López, A.; Arregi, J. M., Conjugacy classes in Sylow \(p- subgroups\) of \(GL (n, q) I\), J. Algebra, 152, 1-19 (1992) · Zbl 0777.20015
[12] Vera-López, A.; Arregi, J. M., Conjugacy classes in Sylow \(p- subgroups\) of \(GL (n, q)\) II, Glasgow Math. J., 36, 91-96 (1994) · Zbl 0810.20039
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