Characterization of a family of generalized companion matrices

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• 作者：C. Garnett^a ; ^{Colin.Garnett@bhsu.edu" class="auth_mail" title="E-mail the corresponding author} ; B.L. Shader^b ; ^{bshader@uwyo.edu" class="auth_mail" title="E-mail the corresponding author} ; C.L. Shader^b ; ^{chan@uwyo.edu" class="auth_mail" title="E-mail the corresponding author} ; P. van den Driessche^c ; ^{pvdd@math.uvic.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11C20 ; 05C20 ; 05C50 ; 15A21 ; 15A54
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：498
• 期：Complete
• 页码：360-365
• 全文大小：247 K

文摘

Matrices A of order n   having entries in the field 5004450&_mathId=si1.gif&_user=111111111&_pii=S0024379515004450&_rdoc=1&_issn=00243795&md5=e7a2fe68bd0249bdb192363a030d6548" title="Click to view the MathML source">F(x₁,…,x_n)$\mathbb{F}(x_1,\dots ,x_n)$ of rational functions over a field 5004450&_mathId=si2.gif&_user=111111111&_pii=S0024379515004450&_rdoc=1&_issn=00243795&md5=9af169825594e113815ee69c25442fd0" title="Click to view the MathML source">F$\mathbb{F}$ and characteristic polynomial are studied. It is known that such matrices are irreducible and have at least 5004450&_mathId=si4.gif&_user=111111111&_pii=S0024379515004450&_rdoc=1&_issn=00243795&md5=c64aed44d165a1f70e6cf092fc23fabb" title="Click to view the MathML source">2n−1$2n-1$ nonzero entries. Such matrices with exactly 5004450&_mathId=si4.gif&_user=111111111&_pii=S0024379515004450&_rdoc=1&_issn=00243795&md5=c64aed44d165a1f70e6cf092fc23fabb" title="Click to view the MathML source">2n−1$2n-1$ nonzero entries are called Ma–Zhan matrices. Conditions are given that imply that a Ma–Zhan matrix is similar via a monomial matrix to a generalized companion matrix (that is, a lower Hessenberg matrix with ones on its superdiagonal, and exactly one nonzero entry in each of its subdiagonals). Via the Ax–Grothendieck Theorem (respectively, its analog for the reals) these conditions are shown to hold for a family of matrices whose entries are complex (respectively, real) polynomials.