Simultaneous triangularization of matrices—low rank cases and the nonderogatory case
Simultaneous triangularization of matrices—low rank cases and the nonderogatory case
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矩阵的联立三角化——低阶情况和非贬义情况
DOI：
10.1080/03081087808817249
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发表时间：
1978
影响因子：
1.1
通讯作者：
T. Laffey
中科院分区：
文献类型：
--
作者：
T. Laffey
g~~ er.. CQEC~~ TI.~~ 的 nrahjem ~ 在 pa11 u~(~ umpk~ j ri;< r; ILIZ&I&> A, 5 whkh dl=> uiZ< kui 2" k19;~ ti, & th 5~; i&E FiGPCi-ty r, 上获得 ronri_irions 的“smaij”ser，即ihar 存在一个非奇异矩阵 T，使得 T-lAT、P'ST Elre 都（上）trian&ar。论文是 &vide& kt G 固定，rp: ectioEs. variass rPrsjts 与 Schneider 的结果相关，如果 4B= 0，则 A、B 具有属性 P，特别是，如果 {fi% 则得到 rhar。 jj2=(Bj. 49"= 6,(t, j= i, L,..., nj，则 A, B 具有属性 B，并且如果秩 (AB-BA)= 1，这也成立。在第 2 节中，表明如果 A、B 具有属性 L 并且 A 具有秩 1，则 A.B 具有属性 P。第 3 节处理低秩幂零矩阵从属性 L 到属性 P 的转换，并与 Taussky 的最新结果相关（J. Algebra 20 (1972), 271-283；I1 Lin. and Multil. Alg. 2 (1974), 195-202）。本节的主要结果是，如果 n< 5 且 AS= B3=(A+ AB)'=(AB)'= 0 对于所有 A，则 A、B 具有属性 P。在第 4 节中，一组九个条件是必要且充分的，以便获得了一对具有属性 P 的 4 x 4 矩阵，在第 5 节中，一个简短的证明是 Williamson 非常好的理论的 ohtaimrl (Amer.= 7=-Math. 57,(2935), 281-2933，表明 d (Aj (AB--BAj for ali divisorsd (A 的最小多项式的 xj 是必要的并且如果 A 是非贬义的，则 A、B 具有性质 P 就足够了，本文给出了几个例子来表明结果在某些意义上是最好的。
g~~ er.. CQEC~~ TI.~~ with the nrahjem of~ btaining a" smaij" ser of ronri_irions on a pa11 u~(~ umpk~ j ri;< r; ILIZ&I&> A, 5 whkh dl=> uiZ< kui 2" k19;~ ti, & th 5~; i&E FiGPCi-ty r, ie ihar there exist a nonsingular matrix T such that T-lAT, P'ST Elre both (upper) trian&ar. The paper is &vide& kt G fix, rp: ectioEs. Se&o= 1, variass rPrsjts related ta a result of Schneider that if. 4B= 0, then A, B have property P, are obtained. In particular,.-it is shown rhar if {fi% jj2=(Bj. 49"= 6,(t, j= i, L,..., nj, then A, B havc property B and that this also holds if rank (AB-BA)= 1. In Section 2, it is shown that if A, B have property L and A has rank 1, then A. B have property P. Section 3 deals with the transition from property L to property P for nilpotent matrices of low rank and is related to recent results of Taussky (J. Algebra 20 (1972), 271-283; I1 Lin. and Multil. Alg. 2 (1974), 195-202).The main result in this section is that if n< 5 and AS= B3=(A+ AB)'=(AB)'= 0 for all A, then A, B have property P. In Section 4 a set of nine conditions which are necessary and sufficient in order that a pair of 4 x 4 matrices have properiy P are obtained, in Sectioil 5, a short proof is ohtaimrl of Williamson's very nice theorern (Amer.= 7=-Math. 57,(2935), 281-2933, showing that the niipotency of d (Aj (AB--BAj for ali divisorsd (xj of the minimal polynomial of A is necessary and sufficient for A, B to have property P if A is nonderogatory. T5u'onghout the paper several examples are given to show that the resuIts are in certain senses best possible.