Constructing strong ℓan>-ifications from dual minimal bases

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• 作者：Fernando De Ter&aacute ; n^a ; ^{fteran@math.uc3m.es" class="auth_mail" title="E-mail the corresponding author} ; Froil&aacute ; n M. Dopico^a ; ^{dopico@math.uc3m.es" class="auth_mail" title="E-mail the corresponding author} ; Paul Van Dooren^b ; ^{paul.vandooren@uclouvain.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：65F15 ; 15A18 ; 14A21 ; 15A22 ; 15A54 ; 93B18
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：495
• 期：Complete
• 页码：344-372
• 全文大小：531 K

文摘

We provide an algorithm for constructing strong ℓ  -ifications of a given matrix polynomial an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000495&_mathId=si1.gif&_user=111111111&_pii=S0024379516000495&_rdoc=1&_issn=00243795&md5=e3b0d235afc5585a8d2068c11ee3c9cc" title="Click to view the MathML source">P(λ)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">Palse">(λalse">)ath>an>an>an> of degree d   and size an id="mmlsi171" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000495&_mathId=si171.gif&_user=111111111&_pii=S0024379516000495&_rdoc=1&_issn=00243795&md5=b189ade333624a1a62153763f7e9e1ee" title="Click to view the MathML source">m×nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si171.gif" overflow="scroll">m×nath>an>an>an> using only the coefficients of the polynomial and the solution of linear systems of equations. A strong ℓ  -ification of an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000495&_mathId=si1.gif&_user=111111111&_pii=S0024379516000495&_rdoc=1&_issn=00243795&md5=e3b0d235afc5585a8d2068c11ee3c9cc" title="Click to view the MathML source">P(λ)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">Palse">(λalse">)ath>an>an>an> is a matrix polynomial of degree ℓ   having the same finite and infinite elementary divisors, and the same numbers of left and right minimal indices as the original matrix polynomial an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000495&_mathId=si1.gif&_user=111111111&_pii=S0024379516000495&_rdoc=1&_issn=00243795&md5=e3b0d235afc5585a8d2068c11ee3c9cc" title="Click to view the MathML source">P(λ)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">Palse">(λalse">)ath>an>an>an>. All explicit constructions of strong ℓ-ifications introduced so far in the literature have been limited to the case where ℓ divides d, though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible. Based on recent developments on dual polynomial minimal bases, we present a general construction of strong ℓ-ifications for wider choices of the degree ℓ, namely, when ℓ divides one of nd or md   (and an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000495&_mathId=si4.gif&_user=111111111&_pii=S0024379516000495&_rdoc=1&_issn=00243795&md5=55a9258831d93ea7846830a5efcf000a" title="Click to view the MathML source">d≥ℓan>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">d≥ℓath>an>an>an>). In the case where ℓ divides nd (respectively, md), the strong ℓ  -ifications we construct allow us to easily recover the minimal indices of an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000495&_mathId=si1.gif&_user=111111111&_pii=S0024379516000495&_rdoc=1&_issn=00243795&md5=e3b0d235afc5585a8d2068c11ee3c9cc" title="Click to view the MathML source">P(λ)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">Palse">(λalse">)ath>an>an>an>. In particular, we show that they preserve the left (resp., right) minimal indices of an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000495&_mathId=si1.gif&_user=111111111&_pii=S0024379516000495&_rdoc=1&_issn=00243795&md5=e3b0d235afc5585a8d2068c11ee3c9cc" title="Click to view the MathML source">P(λ)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">Palse">(λalse">)ath>an>an>an>, and the right (resp., left) minimal indices of the ℓ  -ification are the ones of an id="mmlsi1" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000495&_mathId=si1.gif&_user=111111111&_pii=S0024379516000495&_rdoc=1&_issn=00243795&md5=e3b0d235afc5585a8d2068c11ee3c9cc" title="Click to view the MathML source">P(λ)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">Palse">(λalse">)ath>an>an>an> increased by an id="mmlsi183" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516000495&_mathId=si183.gif&_user=111111111&_pii=S0024379516000495&_rdoc=1&_issn=00243795&md5=5e6aab0893d9a0d412a14521ac95f430" title="Click to view the MathML source">d−ℓan>an class="mathContainer hidden">an class="mathCode">ath altimg="si183.gif" overflow="scroll">d−ℓath>an>an>an> (each). Moreover, in the particular case ℓ divides d, the new method provides a companion ℓ-ification that resembles very much the companion ℓ-ifications already known in the literature.