🚀 go-pugleaf

$BEgED!w:eBg$G$9!%(B

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In article <SHIMADA.96Oct4144317@slson5.cv.ccm.eng.osaka-u.ac.jp> shimada@cv.ccm.eng.osaka-u.ac.jp (Nobutaka Shimada) writes:
>>   $B$"$k9TNs$N9TNs<0$NF3=P$GG:$s$G$$$^$9!%$*CN7C$r$*B_$72<$5$$!%(B
>>
>>	   det| I - qPXXt / ( 1 + qXtPX ) | = 1 / ( 1 + qXtPX ) - (*)
>>	   det| In - qPX( Im + qXtPX )^(-1)Xt |  - (**)

$B$K$D$$$F!$%a!<%k$G$H$"$kJ}$+$i2rEz(B($B0F(B)$B$rAw$C$FD:$-$^$7$?!%$I$&$b(B
$B$"$j$,$H$&$4$6$$$^$7$?!%(B
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$B$^$:!$(B(*)$B$N<0$N:8JU$+$i1&JU$NF3=P$K$D$$$F!'(B

	det| I - qPXXt / ( 1 + qXtPX ) | = 1 / ( 1 + qXtPX )    $B!D(B(1)

$B$3$3$G!"9TNs(BP$B$O0J2<$NMM$KJ,2r$G$-$k(B

P=Qt D Q
Q:$BD>8r9TNs(B      D:$B@5CMBP3Q9TNs(B

$B$3$N$?$a<0(B(1)$B$O(B Y=QX$B$H$*$/$H!$(BI=QtQ$B$rMxMQ$7$F<!<0$KJQ7A$G$-$k!%(B

	det| I - qDYYt / ( 1 + qYtDY ) | = 1 / ( 1 + qYtDY )    $B!D(B(2)

$B$3$3$G!"?7$?$KJQ?t#z!J":#R(B)$B$r0J2<$N$h$&$KDj$a$k(B

	z = q / ( 1 + qYtDY )    $B!D(B(3)

$B$3$N$?$a(B  (2) $B$O0J2<$N$h$&$KJQ7A$G$-$k(B

det| I - zDYYt | = 1 - zYtDY       $B!D(B(4)

$B$3$N(B(4)$B$r<($;$PNI$$!#0J2<(B(4)$B$NEy<0$r@bL@$9$k!#(B
$B$3$3$G(B(4)$B$N:8JU$O0J2<$N<0$GE83+$G$-$k!%(B

1 - z tr(DYYt) + z^2 $B&2(B (DYYt)$B$N#2<!$N<g:B>.9TNs<0(B
               - z^3 $B&2(B (DYYt)$B$N#3<!$N<g:B>.9TNs<0(B $B!\(B $B!D(B  
          $B!D(B  (-z)^n det(DYYt)                          $B!D(B(5)

$B$3$3$G!"(B(DYYt)$B$N#2<!$+$i(Bn-1$B<!$N<g:B>.9TNs<0$O#0$G$"$j!"$^$?(Bdet(DYYt)=0
$B$J$N$G!"<0(B(5)$B$O(B 1 - z tr(DYYt) $B$H$J$k!#(B
$B$3$3$G!"(Btr(DYYt)$B$O(B YtDY$B$H$J$k$N$G<0(B(4)$B$,@.N)$9$k!#(B
$B!J0J>e!K(B

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$B;d<+?H$G$O(B(5)$B<0$NE83+$r$^$@$-$A$s$H%U%)%m!<$G$-$F$$$J$$$N$G$9$,!$(B
$B%i%W%i%9$NE83+$H$$$&E[$G$7$g$&$+!%(B

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$B<!$K(B(**)$B$N>l9g$K$D$$$F!%(B

$B7kO@!'(B
	det| In - qPX( Im + qXtPX )^(-1)Xt | = det | Im + qXtPX | ^(-1)

$B>r7o!'(BX$B$,%U%k%i%s%/$G$"$k$3$H!#(Bn>=m$B$G$"$k$3$H!#(BP$B$,@5CMBP>N9TNs$G$"$k$3$H!#(B

$B>ZL@!'(B

P^(1/2)$B$r<!$N$h$&$KDj5A$9$k!#(B
	P=Qt diag($B&K(B1,$B!D(B,$B&K(Bn) Q $B$HJ,2r$G$-$k!#(B
  	$B$3$N$H$-(B P^(1/2)=Qt diag( $B"e&K(B1 ,$B!D(B, $B"e&K(Bn ) Q $B$H$9$k!#(B

$B$3$3$G(B
	Y$B$r(BP^(1/2)X    $B!D(B(A)

$B$HCV$/!#(B(P^(1/2))t = P^(1/2)$B$G$"$k!J(B(.)t$B$OE>CV$N0U!K!%(B
$B$h$C$F!$(B

	In - qPX( Im + qXtPX )^(-1)Xt =
		P^(1/2) { In - qY(Im + qYtY )^(-1)Yt } P^(-1/2)

$B$H$J$k$N$G(B

	det| In - qPX( Im + qXtPX )^(-1)Xt | =
		det | In - qY( Im + qYtY )^(-1)Yt |  $B!D(B(6)

$B$H$J$k!#(B
$B$^$?(BX$B$,%U%k%i%s%/$J$N$G(BY$B$b%U%k%i%s%/$G$"$k(B(rank(Y)=m)$B!%$h$C$F(B

	     ( D ) 
	RY = ( 0 )	D:$B#m<!@5B'9TNs!$(B0:(n-m)$B!_(Bm$B$NNm9TNs(B $B!D(B(B)

$B$H$J$k(Bn$B<!D>8r9TNs(BR$B$,B8:_$9$k!J1&JU$O(BD,0$B$r=D$K$J$i$Y$?9TNs$r0UL#$9$k!K!%(B
$B$3$N$?$a0J2<$N<0$,@.N)$9$k!#(B
	
	In - qY( Im + qYtY )^(-1)Yt = 
		Rt[ In - q( D ) { Im + q( Dt 0 )( D ) }^(-1)( Dt 0 ) ]R	
		          ( 0 )                 ( 0 )
					$B!D(B(7)

$B$^$?(B(7)$B<01&JU$O<!<0$N$h$&$KJQ7A$G$-$k!#(B
   
	Rt { In  -  ( qD( Im + qDtD )^(-1)Dt     0 ) } R	   $B!D(B(8)
	   {	    (       0                    0 ) }


$B$3$N$?$a(B(6)$B1&JU$N9TNs<0$O<!<0$H$J$k!#(B

	det | Im - qD( Im + qDtD )^(-1)Dt |		$B!D(B(9)

$B$3$3$G!"9TNs(BT,Z$B$r<!$N$h$&$KCV$/(B

	T = Im - qD( Im + qDtD )^(-1)Dt			$B!D(B(10)
	Z = (Im + qDtD)^(-1)				$B!D(B(11)

$B<0(B(10)(11)$B$h$j(B 

	TD = ( D - qDZDtD ) = D - DZ( Z^(-1) - Im ) = DZ	$B!D(B(12)
	($B"h(B qDtD = Z^(-1) - Im, q$B$O%9%+%i!<(B)

$B$G$"$k!%$3$3$G!"(BD$B$O@5B'$G$"$k$3$H$H(B(12)$B$h$j(B

	det|(12)$B:8JU(B| = det|TD| = det|T| det|D|,
	det|(12)$B1&JU(B| = det|DZ| = det|D| det|Z|

	$B!h(B (9)$B<0:8JU(B = det|T| = det|Z| = 1 / det| Im + qDtD |

$B$,@.N)$9$k!%(B
$B$^$?(B(A),(B)$B<0$rMQ$$$k$H!$(B

	XtPX = YtY = ( Dt 0 )RRt( D ) = DtD
	                        ( 0 )

$B$G$"$k$+$i!$(B(9)$B<0:8JU(B = 1 / det| Im + qXtPX | $B$,@.N)$9$k(B
($B0J>e!K(B

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        $BEgED(B $B?-7I!w@)5!!%9)!%:eBg(B $B#D#3(B  06-877-5111 $BFb@~(B3373
     $B5!3#7O#DEo#4#F(B    E-mail  shimada@cv.ccm.eng.osaka-u.ac.jp
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