| 某些矩阵秩不等式的边界条件 |
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| 引用本文： | 刘英.某些矩阵秩不等式的边界条件[J].高师理科学刊,2010,30(2):31-34,49. |
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| 作者姓名： | 刘英 |
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| 作者单位： | 哈尔滨师范大学,恒星学院,信息科学系,黑龙江,哈尔滨,150025 |
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| 基金项目： | 黑龙江省高教学会“十一五”规划项目(115C-580) |
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| 摘 要： | 对几个常见的矩阵秩不等式,讨论其等号成立的条件,并将矩阵和的秩不等式加以细化.得到主要结论:(i)r((A1,,At))=r(Ai)(1≤i≤t)当且仅当有矩阵B与C适合Ai=BA1Ai=AiAtC;(ii)Sylvester不等式r(AB)≥r(A)+r(B)-n中等式成立,当且仅当k≥n-r(k为B的列数,r=r(A),当A=P(Ir0)Q时,B=Q-1(CIn-r)R(P,Q,R为可逆矩阵);(iii)max{r((A,B))-n,r((AB))-m}≤r(A+B)≤min{r((A,B)),r(AB))},(A,B为m×n矩阵),且刻画了等式成立的条件.
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| 关 键 词： | 矩阵 秩 不等式 |
The boundary conditions for some rank inequalities of matrices |
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| LIU Ying.The boundary conditions for some rank inequalities of matrices[J].Journal of Science of Teachers＇College and University,2010,30(2):31-34,49. |
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| Authors: | LIU Ying |
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| Institution: | Department of Information Science;Star College;Harbin Normal University;Harbin 150025;China |
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| Abstract: | Discussed the conditions for the establishment of several common matrix rank inequality, and refined the rank inequality of matrices addition. The main conclusions are （ i ） r（（A1, …, At ）） = r（Ai ） （1 ≤ i ≤ t） if and only if there is matrixB withC satisfying： Ai = BA1 …Ai = Ai …ARC ; （ ii ） Equality in Sylvester＇s inequality sets up r（AB） ≥ r（A） ＋ r（B） - n if and only ifk ≥ n - r （ k is the columns number of B , r = r（A）, when A =P〔^Ir 0〕Q,B=Q^-1〔^C In-r〕R（P,Q,R are invertible matrices ）; （ iii ） max {r（（A,B））-n,r〔〔^AB〕〕-m}≤r（A＋B）≤MIN{R（（A,B））,R〔〔^AB〕〕}（A,B are m × n matrices ） and characterized the conditions for the establishment of the equation. |
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| Keywords: | matrix rank inequalities |
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