LetF be an infinite field and letn,p ₁,p ₂,p ₃ be positive integers such thatn =p ₁ +p ₂ +p ₃. Let\(C_{1,2} \in F^{p_1 \times p_2 } \),\(C_{1,3} \in F^{p_1 \times p_3 } \) and\(C_{2,1} \in F^{p_2 \times p_1 } \). In this paper we show that appart from an exception, there always exist\(C_{1,1} \in F^{p_1 \times p_1 } \),\(C_{2,2} \in F^{p_2 \times p_2 } \) and\(C_{2,3} \in F^{p_2 \times p_3 } \) such that the pair $$(A_1 , A_2 ) = \left( {\left[ {\begin{array}{*{20}c} {C_{1,1} } \\ {C_{2,1} } \\ \end{array} \begin{array}{*{20}c} {C_{1,2} } \\ {C_{2,2} } \\ \end{array} } \right],\left[ {\begin{array}{*{20}c} {C_{1,3} } \\ {C_{2,3} } \\ \end{array} } \right]} \right)$$ is completely controllable. In other words, we study the possibility of the linear system χ (t) =A _1χ(t) +A _2ζ(t) being completely controllable, whenC _1,2,C _1,3 andC _2,1 are prescribed and the other blocks are unknown. We also describe the possible characteristic polynomials of a partitioned matrix of the form $$C = \left[ {\begin{array}{*{20}c} {C_{1,1} } \\ {C_{2,1} } \\ {C_{3,1} } \\ \end{array} \begin{array}{*{20}c} {C_{1,2} } \\ {C_{2,2} } \\ {C_{3,2} } \\ \end{array} \begin{array}{*{20}c} {C_{1,3} } \\ {C_{2,3} } \\ {C_{3,3} } \\ \end{array} } \right] \in F^{n \times n} ,$$ whereC _1,1,C _2,2,C _3,3 are square submatrices (not necessarily with the same size), whenC _1,2,C _1,3 andC _2,1 are fixed and the other blocks vary.

中文翻译：

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部分处方矩阵的可控性

令F为无限大，令n，p ₁，p ₂，p ₃为正整数，使得n = p ₁ + p ₂ + p ₃。设\（C_ {1,2} \ in F ^ {p_1 \ times p_2} \），\（C_ {1,3} \ in F ^ {p_1 \ times p_3} \）和\（C_ {2,1 } \ in F ^ {p_2 \ times p_1} \）。在本文中，我们证明了从例外情况来看，appart始终存在\（C_ {1,1} \在F ^ {p_1 \ times p_1} \），\（C_ {2,2} \在F ^ {p_2 \ p_2} \）和\（C_ {2,3} \ in F ^ {p_2 \ times p_3} \）这样对 $$（A_1，A_2）= \ left（{\ left [{\ begin {array} {* {20} c} {C_ {1,1}} \\ {C_ {2,1}} \\ \ end {array} \ begin {array} {* {20} c} {C_ {1,2}} \\ {C_ {2,2}} \\ \ end {array}} \ right]，\左[{\ begin {array} {* {20} c} {C_ {1,3}} \\ {C_ {2,3}} \\ \ end {array}} \ right]} \ right）$$ 是完全可控的 换句话说，我们研究了线性系统χ（的可能性吨）=阿 _1χ（吨）+甲 _2ζ（吨）是完全可控，当Ç _1,2，Ç _1,3和C ^ _2,1规定，其他块是未知的。我们还描述了形式为 $$ C = \ left [{\ begin {array} {* {20} c} {C_ {1,1}} \\ {C_ {2,1}的分区矩阵的可能特征多项式}} \\ {C_ {3,1}} \\ \ end {array} \ begin {array} {* {20} c} {C_ {1,2}} \\ {C_ {2,2}} \ \ {C_ {3,2}} \\ \ end {array} \ begin {array} {* {20} c} {C_ {1,3}} \\ {C_ {2,3}} \\ {C_ {3,3}} \\ \ end {array}} \ right] \ in F ^ {n \ times n}，$$ 其中C _1,1，C _2,2，C _3,3是平方子矩阵（不是当C _1,2，C _1,3和C _2,1固定且其他块变化时。