Shifted normal forms of polynomial matrices
Bernhard Beckermann, George Labahn, Gilles Villard · 1999
In t,his paper we st,ucly the problen ~ of transforrniug, via in-vertible colu1tln opcrat.ious ~ it matrix polyioruial into a varicty Of.shiftcd forms. Esarnplcs of forms c:overed in out frmwa-ork include a colunm rctluccd form: il triangular fornlz R I%!rInite IlOrInd fOrll1 or it Popov IlorIllal fOrIll alollg wit,11 their shifted courltcrpart,s. I3y obt.aiuiug tlcgrvc bounds for uuiniodiilar niiill,iplicrs of shifted Popor fornis we are able t,o c11lbct1 tlic probleni of conqmtiug il normal forni into 0Iic of deterniiJIing a sliift.ctl forni Of R InininIal pOl~IlOIIliill hiISiS fur all XWXiiltWl IIliLtris polyioruial. Shifted niiuind polynomial lXPX?S cm be conlpu1.d via sigma bases [2! 31 iIlld ill POpOv forui Vii1 Mahler s~sbenls [il. Tl ic 1. d t,t, cr Iut:t.lIotl gives a fractiorl-frw algorithm for computing niatris riormal forms. Key words: Popov Form. Herr&c N~~r~n~d Fornl 1