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thematicsroy

гæ (СÓÐÃûÆø)

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1¡¢ÒÑÖªH1ÊÇÕý¶¨¾ØÕó£¬H2ÊÇHermite¾ØÕó£¬ÇÒÓëH1ͬ½×£¬ÇóÖ¤£º
¡°H1+H2Õý¶¨¡±µÈ¼ÛÓÚ¡°(inv(H1))*H2µÄËùÓÐÌØÕ÷¸ù´óÓÚ-1¡±
2¡¢¾ØÕóPx,Py·Ö±ðÊÇC^nÉÏÓ³Éäµ½¿Õ¼äX£¬YµÄÕý½»Í¶Ó°£¬X,YͬάÊý£¬ÇóÖ¤£º
||Px-Py||=||(I-Px)Py||=||(I-Py)Px||,
ÕâÀï||.||Ϊ¶þ·¶Êý

¾À½áÁ˺þ㬹ÊÀ´ÇóÖúÓÚ¸÷λ£¬Ð»Ð»ÁË~
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» ±¾Ö÷ÌâÏà¹Ø¼ÛÖµÌùÍÆ¼ö£¬¶ÔÄúͬÑùÓаïÖú:

ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû

hank612

ÖÁ×ðľ³æ (ÖøÃûдÊÖ)

¡¾´ð°¸¡¿Ó¦Öú»ØÌû

¸Ðл²ÎÓ룬ӦÖúÖ¸Êý +1
µ±M¶Ô³Æ(»òHermitian), NÕý¶¨Ê±, ¶þÕßÊÇ¿ÉÒÔͬʱ(¶ÔºÏÒâÒåÏÂ) ¶Ô½Ç»¯µÄ.
Äã¿ÉÒÔ¿¼ÂÇNµÄCholesky decomposition, N=L L^H. ÄÇô¶ÔHermitian ¾ØÕó (L^(-1) M {L^(-1)}^H. ¿ÉÒÔÈ¡µ½µ¥Î»Õý½»»ù Y , Ò²¾ÍÊÇ L^{-1} M {L^{-1}}^H Y = Y  Lambda, ÇÒ Y^H Y=I. ÄÇô X=  {L^{-1}}^H Y Âú×ãÒªÇóX^H M X= Lambda, X^H N X = I.
ÏÖÔڻص½ÎÊÌâ1.  M+N Õý¶¨ µ±ÇÒ½öµ± ¶ÔÓ¦±ê×¼¶þ´ÎÐÍϵÊýÈ«Õý: x1^2 +...+xn^2.
Ò²¾ÍÊÇ˵ ÔÚX¶ÔºÏÏ Lambda + I > 0.  µ«ÊÇ Lambda =(X^H N X )^(-1) X^H M X=X^{-1} N^{-1} M X¾ÍÊÇN^{-1} M µÄÌØÕ÷Öµ.

ÎÊÌâ2.  È¡×Ó¿Õ¼ä X= (1, 0), Y=(0,1), ´Ó¶ø PX= ( 1, 0; 0, 0), PY= (0, 0; 0, 1).  ´ÓÓ³Éä½Ç¶È¾ÍÊÇ PX (a,b)=(a,0), PY(a,b)=(0,b).
PX-PY = (1, 0; 0, -1) µ«ÊÇ (I-PX) PY = (0, 0; 0, 1) * (0,0; 0, 1) = (0, 0; 0, 1). ·¶ÊýºÃÏó²»Í¬.
We_must_know. We_will_know.
2Â¥2013-09-18 02:50:32
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû

hank612

ÖÁ×ðľ³æ (ÖøÃûдÊÖ)

ÒýÓûØÌû:
2Â¥: Originally posted by hank612 at 2013-09-18 02:50:32
µ±M¶Ô³Æ(»òHermitian), NÕý¶¨Ê±, ¶þÕßÊÇ¿ÉÒÔͬʱ(¶ÔºÏÒâÒåÏÂ) ¶Ô½Ç»¯µÄ.
Äã¿ÉÒÔ¿¼ÂÇNµÄCholesky decomposition, N=L L^H. ÄÇô¶ÔHermitian ¾ØÕó (L^(-1) M {L^(-1)}^H. ¿ÉÒÔÈ¡µ½µ¥Î»Õý½»»ù Y , Ò²¾ÍÊÇ L^{-1} M { ...

ÎÒ˵µÄ·¶Êý²»Í¬Ö¸µÄÊÇFrobenius¶þ·¶Êý, http://en.wikipedia.org/wiki/Matrix_norm.

Èç¹ûÊǾØÕó¶þ·¶Êý, Max |Ax|/|x|, ÄÇô, ¼ÙÉèX ²»µÈÓÚY, ¿¼ÂÇÏòÁ¿yÊôÓÚY²¢ÇÒÕý½»ÓÚX, ¿ÉÒÔ¿´µ½ÄÇЩ·¶Êý¶¼µÈÓÚ1, Òò´ËÏàµÈ.
We_must_know. We_will_know.
3Â¥2013-09-18 03:06:04
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû

hank612

ÖÁ×ðľ³æ (ÖøÃûдÊÖ)

ÒýÓûØÌû:
3Â¥: Originally posted by hank612 at 2013-09-18 03:06:04
ÎÒ˵µÄ·¶Êý²»Í¬Ö¸µÄÊÇFrobenius¶þ·¶Êý, http://en.wikipedia.org/wiki/Matrix_norm.

Èç¹ûÊǾØÕó¶þ·¶Êý, Max |Ax|/|x|, ÄÇô, ¼ÙÉèX ²»µÈÓÚY, ¿¼ÂÇÏòÁ¿yÊôÓÚY²¢ÇÒÕý½»ÓÚX, ¿ÉÒÔ¿´µ½ÄÇЩ·¶Êý¶¼µÈÓÚ1, Òò´ËÏàµÈ....

my answer is not true for the 2nd problem.
The reason is that there might not exist a non-zero vector which is in Y and also is perpendicular to X.
We_must_know. We_will_know.
4Â¥2013-09-18 09:44:35
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû

hank612

ÖÁ×ðľ³æ (ÖøÃûдÊÖ)

¡ï ¡ï
feixiaolin: ½ð±Ò+2, ½±ÀøÓ¦Öú 2013-12-14 07:25:40
1) ÒòΪ-(P-Q) = (I-P)Q - P(1-Q) ²¢ÇÒ Image (P) ºÍ Image(I-P) Õý½», ËùÒÔ¶ÔÈκε¥Î»ÏòÁ¿ u,
||-(P-Q)u|| >= || (I-P)Q u||, ¼´ ||P-Q||_2 >= || (I-P)Q||_2.

2) ¿¼ÂǶþÎ¬Æ½ÃæÉÏÏòÁ¿ x=(cos(theta), sin(theta) ), y=(0, 1). PΪÔÚxÉÏÕý½»Í¶Ó°, QΪÔÚyÉÏÕý½»Í¶Ó°,  ¼´ P=( cos(theta), sin(theta))^T (cos(theta), sin(theta))= x^T  x,  Q=(0 0; 0 1). Ö±½Ó¼ÆËãµÃµ½
(P-Q) (cos(beta), sin(beta))^T = cos(theta) * (cos(theta-beta), sin(theta-beta))^T,
(I-P)Q (cos(beta), sin(beta))^T = cos(theta)*sin(beta)*(-sin(theta), cos(theta))^T.
´Ëʱ||P-Q||_2 = || (I-P)Q||_2.           ( =cos(theta) )

3) Éè P-QµÄ¶þ·¶ÊýÔÚµ¥Î»ÏòÁ¿uÈ¡µ½, ¼´||P-Q||_2 = ||(P-Q)u||. ¿¼ÂÇu ÔÚPÉÏͶӰx=P(u), ÔÚQÉÏͶӰy=Q(u), ¿ÉÒÔ¿´³ö, uÔÚx ºÍyÉú³ÉµÄ¶þÎ¬Æ½ÃæÉÏ. ÓÉ2)µÄ·ÖÎö, (I-P)Q ÔÚ¸ÃÆ½ÃæÉÏ´æÔÚµ¥Î»ÏòÁ¿ v, ʹµÃ||(P-Q)u||= || (I-P)Q v||, Òò´Ë||P-Q||_2 <=  || (I-P)Q||_2.

¿´À´²»ÂÛÊÇ ¶þ ·¶Êý, »¹ÊÇ ¶þ  Î¬Æ½Ãæ, »¹ÊÇÄÜÓÐһЩºÃÐÔÖʵÄ.

ÁíÍâ, PºÍQ άÊýÏàͬÊDZØÒªµÄ, ±£Ö¤ P(u) ºÍ Q(u) ¶¼²»ÎªÁã. Èç¹ûûÓÐÕâ¸öÌõ¼þ, ¿ÉÒԲο¼ P=Diag(0, 1,1), Q=Diag(0,1,0). Õâʱºò P-Q=Diag(0,0,1), (1-P)Q=0.
We_must_know. We_will_know.
5Â¥2013-09-19 01:51:51
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû
6Â¥2013-12-12 19:02:15
ÒÑÔÄ   »Ø¸´´ËÂ¥   ¹Ø×¢TA ¸øTA·¢ÏûÏ¢ ËÍTAºì»¨ TAµÄ»ØÌû
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