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Multilinear mapping and its application to determinants

  • 1. !#$ & " % ' ( " )* !' + &, " -$ . # & % % /& & - ! 0% 1 &$. + - ) * % 2 " !2 &&3 4 .% )& ! " . - -! & # " $ -! 0! ! - "! " - *+ ! ! & 5 ! - &&& -!$ $ & - 6 & %7! - & % 0 % &6 " &, & ! 1 3&$. " % ! & ! + %& & - 41 # ! + 2 & 5+ . " ! + & n & $ && 8 ! -$ " 3 3 -! $ -! 0 & . '$ -& 6 ! )3$ $ ) 5% - 3 % - ! 0 *& ! 3& A [aij ]n n & ! 0 *& 1 ' ! a11 a1n A an1 ann n n 4# -! ! - -!$ - ! 0A & 1 ' & & & det : A &&0 n det A 1 i j aij M ij &&&, !- 5 . '$ - ! ) & i / 6 $ j 1 n 1 i j aij M ij ,!- 5 . '$ - ! % & j /& 6 i 1 ' & M ij & ! -& 5 -! - $ $ & i &$ . % & j & - !+ 3 ) % ! 0A & - ! - .% ) !" -! $ - ! 0 # .1 - 5 &4 2 - 1 3 5& - &$ . % 3 "- 2 5 2
  • 2. 3 & S {1,2, , k} & 0 - 1 ! &k & $ & " !& & - !+ 3 0 & S & ! )& $ " % )& ! " &9:9&$. 1 ' : {1, , k} {1, , k} ' & ! * " % 3 #- 2 & S k &$ 0 - &&3 "% 2 ! - !+ 3 0 & S &$. S k & ! %& & 1 2 k 1 2 k 0. & Sk & ! ) "% 2 ! & k! & # " & $" &3&& S {1,2,3} &&& ! #& &&;" $ "% $ & 2 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 % % % % $. %& 1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1 - 5 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 S3 , , , , , 1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1 * "% % 1 " $ - 1 &' !+ - $ 5 " & 0 3# 1 i j k && & & & 1 j i k &&3 & S k & .% & & & )1 2- %" 1 #% .% & & & &$. )1 2- % " 1 &' - 5 ! - "% & & sgn : S k { 1,1} && 1 ' 1, 1, , k /2 4/ +-0 6 ( 0 35 &, /1' 7 ' 5 . & 6 ) * ) sgn 1, 1, , k /2 4/ +-0 & ( 0 35 &, /1' ) ' 5 . ) * ) *
  • 3. 1 2 3 4 9<&&& sgn 1 2 4 1 3 - 5 2 4 1 3 1: 1 4 2 3 &!1 2- % " " &;& "% 2: 1 2 4 3 3: 1 2 3 4 & 1 2 3 4 5 =<&&& sgn 1 3 5 2 1 4 - 5 3 5 2 1 4 1: 1 5 2 3 4 2: 1 2 5 3 4 &&&&&&&! 1 2- % " " &>& "% # 3: 1 2 3 5 4 4: 1 2 3 4 5 3 + + &)& T : U V & + & 5 1 - + & ?# -& U & ! ?# ! -& V &," 6 - 1 @ % & /& ! )& T & . - % "5 - - 2 89 &&&, ! + " /& & " T u1 u 2 T u1 T u2 :9 &&&, ! - 8 & " /& &&&&&&& T au aT u - *! 5 - 5 &&&&&& T a1u1 a2 u 2 a1T u1 a2T u 2 %&& a1 , a2 % u , u1 , u 2 U . $ 0 - + &U & &V & & L U ;V ) '! - 3 - % ? " * # ! ! - ! ? &' # ! ! + " - 2 ! !. & - 5 & && " !- 2
  • 4. 3 & U1 , , U k &$ . & %& V ?# ! -" 6 & & @ % . & f : U1 Uk V & & &) - % " 5 - - f u1 ,..., aui bui ,..., u k af u1 ,..., ui ,..., u k bf u1 ,..., ui ,..., u k 1 " * & a, b &$.ui , ui %& U i , i 1,..., k && - 5& * + + & 1 " *& A -& ui &&) B .- * + ! & U1 5 - Uk U && ! 3 & U k U ! U &, & 2 % & V /&$ . &&3 2 3 C% ( & + Lk U ; &$ 0 - *+ &U k & & !&&3& f : U k & * + & .% f !! & " &&) f u1 , , uk : f u 1 , ,u k f u1 , , uk &$..% & % * + & f !! & " & 5 -& &) f u1 , , uk : f u 1 , ,u k sgn f u1 , , uk 1 "* & Sk % -- 5 -& * + ! ! &f 5 ! - u1 , , uk & .% 5 ! & ! !+ &=& 3& - 5 - u1 , , u k &! %& " && - 5 f u1 , , ui , ,u j, , uk f u1 , ,u j, , ui , , uk 1 " * & u1 , , uk U &$ .i, j 1, , k &' & i j %& $. % 3 &f 0& . !5 - !+ 3 # 3& U & ! # - 5 % 5 & ui u j & . - &&) f u1 , , ui , , ui , , uk f u1 , , ui , , ui , , uk - 5& f u1 , , ui , , ui , , uk 0 3 2 ? -& ! # -& Lk U ; &&0! !+ & *+ ! ! & $ 5 ! - & Lk U ; &&, a - & & ! )& /&& 5 3 C ( & U , & 5 U & 5 - + % - -& -$ ! $ + . & * & " + *+ ! ! &&3 5 ! - B .! & k 0 && 5 -
  • 5. !&& ; U &, 0 - 1 / 5 - & k 1 & ! 8 U U * &, ? # &0 !+ 3 & U * & & # ! & /&&$. %3 !+ 3 ? && U & ! # && & & "# $ $ % ! 3 & A !A1 , , An " & ! ! 0 $ % !+ . -3 & n & % 5 Ai -& n & 1 " * & i 1,..., n &&& 2 A & & ! 0 * & n n && 1 ' a1i a11 a1n Ai &&&& 2A &&& ani n 1 an1 ann n n !) # ' & -!$ - ! 0A & [aij ]n n& * + ! ! & - 5 ! 5 !. 3 - 1 ! 3&4# -$ - # B . 5A !& - [aij ]3 3 && 1 ' a11 a12 a12 a13 A !A1 A2 A2 A3 " &&& ! && A1 - 5 a21 % A2 a22 % A2 a22 $.A3 %& a23 & a31 a32 a32 a33 det A A 1 '& a11 $a12 #a12 a13 a21 $a22 #a22 a23 a11 $a22 # a22 a33 a23 $a12 #a12 a31 a13 $a32 # a32 a21 a31 $a32 #a32 a33 a31 $a22 # a22 a13 a21 $a12 #a12 a33 a11 $a32 # a32 a23 $ a11a22 a33 $ a23 a12 a31 $ a13 a32 a21 $ a31a22 a13 $ a21a12 a33 $ a11a32 a23 # a11a22 a33 # a23 a12 a31 # a13 a32 a21 # a31a22 a13 # a21a12 a33 # a11a32 a23 < $. % a11 $a12 a13 a11 #a12 a13 a21 $a22 a23 a21 # a22 a23 $ a11a22 a33 $ a23 a12 a31 $ a13 a32 a21 a31 $a32 a33 a31 # a32 a33
  • 6. $ a31a22 a13 $ a21a12 a33 $ a11a32 a23 # a11a22 a33 # a23 a12 a31 # a13 a32 a21 # a31a22 a13 # a21a12 a33 # a11a32 a23 &&&&&,D/ D " ! &,D / " D &,D/&&& 2det : A & & * + - . 3$ & -! $ - ! 0 A &! ! ! ! & ! 3 & & " 5 ! - ! A ! A1 A2 A3 " &&' & A2 A2 A2 &&0 - & A1 , A2 , A2 , A3 & 1 ! &$ . % ! - -! ' && $ !) 3 $ det [ A1 A2 A3 ] det [ A1 A3 A2 ] det [ A2 A1 A3 ] det [ A3 A2 A1 ] $. %&) - 5 ! " % - -!$ " 1 2 3 det [ A1 A2 A3 ] sgn det [ A1 A2 A3 ] 1 det [ A1 A2 A3 ] 1 2 3 1 2 3 det [ A1 A2 A3 ] sgn det [ A1 A3 A2 ] 1 det [ A1 A3 A2 ] 1 3 2 1 2 3 det [ A1 A2 A3 ] sgn det [ A2 A1 A3 ] 1 det [ A2 A1 A3 ] 2 1 3 1 2 3 det [ A1 A2 A3 ] sgn det [ A3 A2 A1 ] 1 det [ A3 A2 A1 ] 3 2 1 2 -! & $ - ! 0A & [aij ]3 3 & * + ! ! - 5 ! & - 2&) ! 5 " " 2E . - -!$ - ! 0A & [aij ]n n &&0 " 89 det [ A1 ,..., $Ai ,..., An ] $ det [ A1 ,..., An ] :9 det [ A1 ,..., A j ,..., Ai ,..., An ] det [ A1 ,..., Ai ,..., A j ,..., An ] ;</ det [ A1 ,..., Ai ,..., Ai ,..., An ] 0& =9 det [ A1 ,..., Ai $A j ,..., An ] det [ A1 ,..., An ] 1 " * & i, j &' & i j >9 det [ A1 ,..., $Ai #A j ,..., An ] $ det [ A1 ,..., Ai ,..., An ] # det [ A1 ,..., A j ,..., An ] 1 " * & i, j $.$ , # %& .8 ! - &F</&' - -" " &9</&) !- * + ! ! & - 5 ! 2
  • 7. 9</&&&) # ! &, 5 )/& ? - $ ! 3 - ! 0A & ? & &$ & . -!$ - ! 0 ! "$ A && - + 3 & $a1 b1 a1 $b1 a1 b1 $a1 $b1 a1 b1 $ $a2 b2 a2 $b2 a2 b2 a2 b2 $a2 $b2 0 ! -2 " B .- ! * + &, # ! &F</& ! - " - 5 3& # 0 =</&&& ! % - ! &, 5 ) /&3A - ! 0 A & . 1 5 " - ? - $ & & 3 -!$ 2 % - ! &' - 2 ! ! ! 5 " - ! - 5 * + -&& - + a1 b1 b1 a1 a1 b1 a2 b2 - 5&&& a2 b2 b2 a2 a2 b2 a1 b1 & ;</&&&) !+ 3 - ! &, 5 )/&3A - ! 0 A & ! $ & ? - $ & & 5 % - -! $ - ! 0A & " # && 5 & - B .- ! ! ! " 5 ! - - * & ! 1 3& + 5 - Ai A j . && det [ A1 ,..., Ai ,..., Ai ,..., An ] det [ A1 ,..., Ai ,..., Ai ,..., An ] && 2 det [ A1 ,..., Ai ,..., Ai ,..., An ] 0 & ></&&& ! ) 1 & - ! &, 5 )/& &$ ? - $ ! 3 - ! 0A & " - ? & !& ? , 5 )/&3 -$ ' -!$ -& A & ! % $ %&&3 2 . # B . 5n !& - 3 &$ . % ! 3& Ai ! 3 , i 1,2,3 det [ A1 $A1 A2 A3 ] det [ A1 $A1 A3 ] det [ A1 A2 A3 ] &&&&&&&&&&&&&&, ! - " &F</ $ det [ A1 A1 A3 ] det [ A1 A2 A3 ] &&&&&&&&&&&&, ! - " &9</& det [ A1 A2 A3 ] &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&, ! - " &;</ - + a1 b1 a1 $b1 b1 a1 $a1 b1 a1 $a2 b1 $b2 a1 b1 a2 b2 a2 $b2 b2 a2 $a2 b2 a2 b2 a2 $a1 b2 $b1 & F</&&& ! - 2 " ! !' - * + 0 $ - $& 1" % & A [aij ]n n & - 4 % # $ " H 3 # - - "G
  •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