Simultaneous reduction to companion and triangular forms of sets of matrices. (English) Zbl 0716.15011
The following three questions about sets of \(m\times m\) complex matrices are investigated:
When does a set of matrices admit simultaneous reduction under similarity to upper triangular forms? When do two sets of matrices admit simultaneous reduction to complementary triangular forms? When does a set of matrices admit simultaneous reduction to companion forms?
The authors point out that the first question has been studied for a long time, that the second one is related to complete factorization of rational matrix functions, and that consideration of how to make results on the first two questions coordinate free leads to the third one.
When does a set of matrices admit simultaneous reduction under similarity to upper triangular forms? When do two sets of matrices admit simultaneous reduction to complementary triangular forms? When does a set of matrices admit simultaneous reduction to companion forms?
The authors point out that the first question has been studied for a long time, that the second one is related to complete factorization of rational matrix functions, and that consideration of how to make results on the first two questions coordinate free leads to the third one.
Reviewer: P.M.Gibson
MSC:
| 15A21 | Canonical forms, reductions, classification |
Keywords:
complex matrices; simultaneous reduction under similarity; upper triangular forms; complementary triangular forms; companion forms; complete factorization; rational matrix functionsReferences:
| [1] | Alpay D., Operator Theory: Advances and Applications |
| [2] | DOI: 10.1016/0024-3795(86)90306-X · Zbl 0612.47013 · doi:10.1016/0024-3795(86)90306-X |
| [3] | DOI: 10.1007/BF01682737 · Zbl 0383.47012 · doi:10.1007/BF01682737 |
| [4] | Bart H., Operator Theory: Advances and Applications 1 (1979) |
| [5] | DOI: 10.1137/0318051 · Zbl 0458.93018 · doi:10.1137/0318051 |
| [6] | DOI: 10.1016/0024-3795(88)90228-5 · Zbl 0645.15012 · doi:10.1016/0024-3795(88)90228-5 |
| [7] | DOI: 10.1016/0024-3795(88)90229-7 · Zbl 0645.15013 · doi:10.1016/0024-3795(88)90229-7 |
| [8] | Bart H., Operator Theory: Advances and Applications |
| [9] | Bart H., Simultaneous companion and triangular forms of pairs of matrices (1988) |
| [10] | DeWilde P., IEEE Trans. Circuits and Systems 22 pp 387– (1975) · doi:10.1109/TCS.1975.1084106 |
| [11] | Gohberg I., Canadian Math. Soc. Series of Monographs and Advanced Texts (1986) |
| [12] | DOI: 10.1080/03081087808817249 · Zbl 0399.15008 · doi:10.1080/03081087808817249 |
| [13] | DOI: 10.1016/0024-3795(86)90311-3 · Zbl 0609.15004 · doi:10.1016/0024-3795(86)90311-3 |
| [14] | Lancaster P., The Theory of Matrices (1985) · Zbl 0578.62099 |
| [15] | DOI: 10.1090/S0002-9904-1936-06372-X · Zbl 0015.05501 · doi:10.1090/S0002-9904-1936-06372-X |
| [16] | Sahnovic L. A., Soviet Math. Dokl. 17 pp 203– (1976) |
| [17] | DOI: 10.2307/2371205 · Zbl 0011.19409 · doi:10.2307/2371205 |
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