feng's Blog

Happy coding

对SVD唯一性的理解

feng posted @ Wed, 17 Oct 2012 00:12:52 +0800 in 未分类 , 12665 readers

 

这篇文章主要讨论一下对Singular value decomposition的理解。
 
SVD告诉我们,对于任何一个$m\times n$的矩阵$A$,都存在这样的一个分解:
\[A=U\Sigma V'\]
其中$U$是$m\times m$的酉矩阵,也就是$UU^* = I$;$V$是一个$n\times n$的酉矩阵;$\Sigma$是一个$m\times n$的矩阵,非对角上的元素都是0,对角线上的元素都是非负的实数,不管$A$是实数矩阵,还是酉空间中的矩阵,或者说有虚数元素。
 
定理告诉了我们总是存在一个这样的分解的,但并不是说这样的分解是唯一的。比如说有一个permutation matrix,$J$。permutation matrix就是把单位矩阵的行进行重排列,或者列进行重排列。比如说把单位矩阵的第一行和第二行进行交换,第四行和第九行进行交换,交换后的矩阵就是一个permutaion matrix。一个矩阵$A$,左乘一个$J$,相当于对$A$的相应行进行行交换。右乘的话,相当于列交换。两个$J$相乘为一个单位矩阵,相当于对单位矩阵交换了两行之后,再交换一次,不变。
 
回到正题,任何一个$J$拿到了之后,我们有下面的式子成立
\[A=(UJ_m)(J_m\Sigma J_n)(VJ_n)'\]
$UJ_m$依然是一个酉矩阵,$VJ_n$也依然是一个酉矩阵。这里$J$的小角标表示$J$的大小,是$m\times m$的还是$n\times n$的。如果说$m\ge n$,并且$J_n$是$J_m$左上角的,$J_m$右下角是一个单位阵,右上角和左下角都是0矩阵的话,那么$J_m\Sigma J_n$依然也是一个对角阵。之前对$J_m$和$J_n$的描述,相当于是说,如果$J_m$使得$\Sigma$的第$i$行和第$j$行交换的话,那么$J_n$应该使得$\Sigma$的第$i$列和第$j$列进行交换。从而使得$J_m\Sigma J_n$依然是一个非对角线上元素都是0的矩阵,相当于是将$\Sigma$的对角线上的元素进行了一个重新排列。这是事实告诉我们,SVD的分解是不唯一的。
 
接下来再考虑一点,如果说$\Sigma$唯一确定之后,这个$U$和$V$是否能够唯一确定的呢?我们再引入一个矩阵,用$K_m$表示,是一个$m\times m$的对角方阵,对角线上的元素具有$e^{i\phi}$的形式,其中$\phi$可是互相不相同。根据定义,我们有$K_mK_m^* = I$。那么
\[A=UK_m K_m^*\Sigma K_n(VK_n)^*\]
其中$U K_m$和$VK_n$分别仍然是酉阵,如果说$K_m$的对角线上第$i$个元素和$K_n$对角线上第$i$个元素相同的话,那么$K_m^*\Sigma K_n = \Sigma$。也就是说,如果$U$的第$i$列乘以$e^{i\phi}$,并且$V$的第$i$列也乘以$e^{i\phi}$,那么结论依然成立。
 
然后,我们继续疑问:如果$\Sigma$给定,并且允许$U$和$V$差一个$e^{i\phi}$的话,那么$U$和$V$是否能够唯一确定?
这一种情况,比较麻烦。如果说$\sigma_i$都是non-degenerate的话,那么是唯一确定的。否则的话,依然不能够唯一确定。
 
对degenerate的定义是这样子的。如果$\sigma_i$是degenerate的,那么它有两个互相独立的singular vector。
在这里补充一下,singular value就是$\sigma_i$,而$\mathbf{u}_i$和$\mathbf{v}_i$分别是singular vector,其中要求$i<=\min{m, n}$。并且有下面的式子成立
\[A\mathbf{v}_i = \sigma \mathbf{u}_i\]
并且
\[A^*\mathbf{u}_i = \sigma \mathbf{v}_i\]
 
如果说$\sigma_i$都是不一样的,那么所有的$\sigma_i$都是non-degenerate的,也就是说这个时候,$U$和$V$都是唯一确定的,在一定意义下。如果说$A$是一个实数矩阵,$U$和$V$可以也是实数的,这样的话,对于其唯一确定的理解是,差一个符号。
 
这里对于degenerate的讨论比较少,以后有时间补上。
Avatar_small
buy facebook comment said:
Tue, 20 Apr 2021 09:02:18 +0800

I love the way you write and share your niche! Very interesting and different! Keep it coming!

Avatar_small
Car Bluetooth Receiv said:
Tue, 27 Apr 2021 03:15:34 +0800

I know your expertise on this. I must say we should have an online discussion on this. Writing only comments will close the discussion straight away! And will restrict the benefits from this information.

Avatar_small
bed set said:
Mon, 04 Oct 2021 20:18:43 +0800

Positive site, where did u come up with the information on this posting? I'm pleased I discovered it though, ill be checking back soon to find out what additional posts you include.

Avatar_small
www.spiw.org said:
Fri, 08 Oct 2021 22:58:13 +0800

Hi! Thanks for the great information you have provided! You have touched on crucuial points!

Avatar_small
credit card processi said:
Sat, 09 Oct 2021 20:13:42 +0800

I love the way you write and share your niche! Very interesting and different! Keep it coming!

Avatar_small
best forex brokers said:
Sun, 24 Oct 2021 16:24:50 +0800

I high appreciate this post. It’s hard to find the good from the bad sometimes, but I think you’ve nailed it! would you mind updating your blog with more information?

Avatar_small
custom home builder said:
Sun, 24 Oct 2021 16:25:04 +0800

I high appreciate this post. It’s hard to find the good from the bad sometimes, but I think you’ve nailed it! would you mind updating your blog with more information?

Avatar_small
AAA said:
Sat, 01 Jan 2022 14:49:03 +0800

Somebody essentially help to make seriously posts I would state. This is the very first time I frequented your website page and thus far? I surprised with the research you made to make this particular publish extraordinary. Excellent job! 안전놀이터

Avatar_small
AAA said:
Thu, 17 Feb 2022 04:57:00 +0800

small gardens are cute but i still prefer to have those medium sized gardens where i can grow some veggies- 토토사이트

Avatar_small
meidir said:
Sat, 03 Dec 2022 18:40:20 +0800

Hiya, My partner and i have a look at all your documents, you can keep them forthcoming. iphone 14

Avatar_small
seo service UK said:
Tue, 16 Jan 2024 16:37:25 +0800

"After reading your article I was amazed. I know that you explain it very well. And I hope that other readers will also experience how I feel after reading your article.

Avatar_small
안전놀이터 said:
Tue, 20 Feb 2024 14:30:51 +0800

Wonderful site. Plenty of helpful information here. I am sending it to a few buddies ans also sharing in delicious. And certainly, thanks in your effort! Woman of Alien Ideal work you might have completed, this website is absolutely interesting with fantastic details. Time is God as way of retaining everything from happening directly. Thank you for helping out, superb information. In case of dissension, never dare to judge till you ave heard the other side. That is a great tip especially to those new to the blogosphere. Short but very accurate information Appreciate your sharing this one. A must read article! very good put up, i definitely love this web site, keep on it

Avatar_small
sm카지노 said:
Tue, 20 Feb 2024 14:31:19 +0800

You really make it seem so easy with your presentation but I find this topic to be really something that I think I would never understand. It seems too complex and extremely broad for me. I’m looking forward for your next post, I’ll try to get the hang of it!

Avatar_small
먹튀검증사이트 said:
Tue, 20 Feb 2024 14:31:47 +0800

Wonderful site. Plenty of helpful information here. I am sending it to a few buddies ans also sharing in delicious. And certainly, thanks in your effort! Woman of Alien Ideal work you might have completed, this website is absolutely interesting with fantastic details. Time is God as way of retaining everything from happening directly. Thank you for helping out, superb information. In case of dissension, never dare to judge till you ave heard the other side. That is a great tip especially to those new to the blogosphere. Short but very accurate information Appreciate your sharing this one. A must read article! very good put up, i definitely love this web site, keep on it

Avatar_small
토토사이트추천 said:
Tue, 20 Feb 2024 14:35:07 +0800

Hey there, I think your blog might be having browser compatibility issues. When I look at your blog in Safari, it looks fine but when opening in Internet Explorer, it has some overlapping. I just wanted to give you a quick heads up! Other then that, awesome blog!

Avatar_small
온라인카지노순위 said:
Tue, 20 Feb 2024 14:35:27 +0800

Excellent blog! Do you have any recommendations for aspiring writers? I’m planning to start my own blog soon but I’m a little lost on everything. Would you recommend starting with a free platform like WordPress or go for a paid option? There are so many options out there that I’m totally confused .. Any ideas? Thanks!

Avatar_small
먹튀폴리스보증업체 said:
Tue, 20 Feb 2024 14:35:52 +0800

Well said. I in no way concept I ought to accept as true with this opinion, but I’m beginning to view topics from a specific view. I actually have to research extra on this as it seems very thrilling. One hassle I don’t understand despite the fact that is how the whole lot is associated collectively.

Avatar_small
안전토토사이트 said:
Tue, 20 Feb 2024 14:59:09 +0800

This is a truly good site post. Not too many people would actually, the way you just did. I am really impressed that there is so much information about this subject that have been uncovered and you’ve done your best, with so much class. If wanted to know more about green smoke reviews, than by all means come in and check our stuff.

Avatar_small
안전놀이터가입 said:
Tue, 20 Feb 2024 14:59:26 +0800

I wish to show thanks to the writer for rescuing me from this challenge. As a result of exploring through the search engines and seeing thoughts which were not powerful, I was thinking my entire life was well over. Existing without the presence of answers to the difficulties you’ve sorted out as a result of your main blog post is a crucial case, as well as ones that might have in a negative way damaged my career if I had not discovered your site. That understanding and kindness in dealing with every aspect was precious. I don’t know what I would have done if I had not come upon such a step like this. I am able to at this time look forward to my future. Thanks for your time very much for your professional and result oriented help. I won’t hesitate to propose your web page to any person who should have guide about this subject matter. Needed to write you that bit of observation to help give thanks the moment again for those splendid opinions you’ve discussed here. It has been so unbelievably generous with people like you to grant easily precisely what many people would have supplied for an ebook to get some profit for themselves, certainly since you could have done it in case you wanted. The solutions additionally served to become a easy way to recognize that some people have the same desire much like my very own to figure out way more with regard to this issue. I’m sure there are lots of more fun occasions ahead for people who view your website.

Avatar_small
먹튀검증사이트 said:
Tue, 20 Feb 2024 14:59:46 +0800

This is a truly good site post. Not too many people would actually, the way you just did. I am really impressed that there is so much information about this subject that have been uncovered and you’ve done your best, with so much class. If wanted to know more about green smoke reviews, than by all means come in and check our stuff.

Avatar_small
토토사이트종류 said:
Tue, 20 Feb 2024 15:05:22 +0800

Needed to send you the little remark to be able to give many thanks yet again on your exceptional tips you’ve discussed on this website. It is simply extremely generous with you to grant extensively what exactly a number of us could have sold for an electronic book to generate some profit on their own, precisely considering that you might well have done it in case you wanted. These good tips likewise worked like the great way to fully grasp that some people have similar passion just like my very own to realize more and more in terms of this matter. I am sure there are a lot more pleasurable situations ahead for many who read through your blog post. I really wanted to develop a brief comment in order to express gratitude to you for all the amazing information you are writing on this site. My long internet investigation has finally been recognized with incredibly good facts to share with my two friends. I would assert that most of us site visitors actually are really lucky to be in a superb network with many special professionals with beneficial tricks. I feel extremely blessed to have come across the web pages and look forward to really more fun minutes reading here. Thanks again for all the details.

Avatar_small
먹튀검증사이트 said:
Tue, 20 Feb 2024 15:27:47 +0800

thanks for taking the time to speak about this, I experience strongly approximately it and love studying greater on this subject matter. If viable, as you benefit know-how, could you mind updating your weblog with extra facts? it's miles extremely useful for me.

Avatar_small
안전놀이터 said:
Tue, 20 Feb 2024 15:28:06 +0800

This is a truly good site post. Not too many people would actually, the way you just did. I am really impressed that there is so much information about this subject that have been uncovered and you’ve done your best, with so much class. If wanted to know more about green smoke reviews, than by all means come in and check our stuff.

Avatar_small
메이저안전놀이터 said:
Tue, 20 Feb 2024 15:28:08 +0800

Needed to send you the little remark to be able to give many thanks yet again on your exceptional tips you’ve discussed on this website. It is simply extremely generous with you to grant extensively what exactly a number of us could have sold for an electronic book to generate some profit on their own, precisely considering that you might well have done it in case you wanted. These good tips likewise worked like the great way to fully grasp that some people have similar passion just like my very own to realize more and more in terms of this matter. I am sure there are a lot more pleasurable situations ahead for many who read through your blog post. I really wanted to develop a brief comment in order to express gratitude to you for all the amazing information you are writing on this site. My long internet investigation has finally been recognized with incredibly good facts to share with my two friends. I would assert that most of us site visitors actually are really lucky to be in a superb network with many special professionals with beneficial tricks. I feel extremely blessed to have come across the web pages and look forward to really more fun minutes reading here. Thanks again for all the details.

Avatar_small
먹튀검증사이트 said:
Tue, 20 Feb 2024 15:28:57 +0800

I really impressed after read this because of some quality work and informative thoughts. I just wanna say thanks to the writer and wish you all the best for coming!. We at Unique Submission. We have our ability in each type of Online Assignment Help which Includes Dissertation Writing, Essay Writing, Assignment composing, Research Proposal Writing, Admission essay composing, Coursework composing and many more. Excellent blog I visit this blog it’s really awesome. The important thing is that in this blog content written clearly and understandable. The content of information is very informative. Thanks for sharing! If you are using a Garmin Express device, an update will offer you newer features because the company always works on developing their older features to give customers a better experience.

Avatar_small
먹튀사이트조회 said:
Tue, 20 Feb 2024 15:29:59 +0800

vI must show some thanks to you just for bailing me out of this type of problem. Because of researching through the search engines and coming across advice which are not productive, I was thinking my entire life was well over. Living without the presence of solutions to the problems you’ve solved through your entire site is a crucial case, as well as the kind which could have in a wrong way damaged my career if I had not encountered your web site. The knowledge and kindness in dealing with all the things was very useful. I don’t know what I would have done if I had not discovered such a solution like this. I am able to at this moment relish my future. Thanks for your time so much for this reliable and effective guide. I won’t hesitate to suggest your web blog to any individual who ought to have tips on this area.

Avatar_small
토토사이트 said:
Tue, 20 Feb 2024 15:30:06 +0800

I and also my friends ended up following the nice thoughts from the blog and so quickly I got a terrible feeling I never expressed respect to the web site owner for those strategies. The women were certainly warmed to read through them and now have pretty much been making the most of them. Appreciate your turning out to be well thoughtful and also for selecting certain important issues most people are really needing to be aware of. Our own honest regret for not expressing appreciation to earlier. Needed to compose you that very little observation so as to say thanks a lot once again about the pleasing methods you have shown in this case. It was so surprisingly generous with you to offer unhampered precisely what some people would’ve sold as an electronic book to earn some profit on their own, precisely given that you could possibly have done it in case you desired. These techniques as well served like the fantastic way to realize that the rest have the identical zeal really like mine to understand a lot more pertaining to this problem. I am certain there are lots of more enjoyable sessions up front for individuals that scan your blog post.

Avatar_small
사설토토 said:
Tue, 20 Feb 2024 15:57:25 +0800

Youre so awesome! I don’t intend Ive read anything similar to this prior to. So wonderful to find someone with some original thoughts on this subject. realy thank you for starting this up. this website is something that is required online, a person with a little creativity. useful job for bringing something brand-new to the net! This is a great inspiring article.I am pretty much pleased with your good work.You put really very helpful information. Keep it up. Keep blogging Looking to reading your next post. Terrific article! This is the kind of info that are meant to be shared around the net. Shame on Google for not positioning this post higher! Come on over and visit my website. Nice blog here! Thanks =)|

Avatar_small
아도에이전트 said:
Tue, 20 Feb 2024 15:57:44 +0800

I and also my friends ended up following the nice thoughts from the blog and so quickly I got a terrible feeling I never expressed respect to the web site owner for those strategies. The women were certainly warmed to read through them and now have pretty much been making the most of them. Appreciate your turning out to be well thoughtful and also for selecting certain important issues most people are really needing to be aware of. Our own honest regret for not expressing appreciation to earlier. Needed to compose you that very little observation so as to say thanks a lot once again about the pleasing methods you have shown in this case. It was so surprisingly generous with you to offer unhampered precisely what some people would’ve sold as an electronic book to earn some profit on their own, precisely given that you could possibly have done it in case you desired. These techniques as well served like the fantastic way to realize that the rest have the identical zeal really like mine to understand a lot more pertaining to this problem. I am certain there are lots of more enjoyable sessions up front for individuals that scan your blog post.

Avatar_small
개구리 said:
Tue, 20 Feb 2024 15:58:00 +0800

An impressive share, I simply offered this onto a colleague that was doing a little evaluation on this. As well as he in fact bought me breakfast due to the fact that I discovered it for him. smile. So let me reword that: Thnx for the treat! But yeah Thnkx for spending the moment to review this, I really feel strongly concerning it and like reading more on this subject. Ideally, as you end up being experience, would certainly you mind upgrading your blog site with more information? It is very helpful for me. Big thumb up for this post! Quality posts is the secret to interest the people to visit the site, that’s what this web page is providing.| Nice post. I learn something totally new and challenging on websites I stumbleupon every day. It’s always interesting to read through articles from other authors and practice a little something from other websites.

Avatar_small
먹튀폴리스도메인 said:
Tue, 20 Feb 2024 16:50:15 +0800

"Hi there! This post couldn’t be written any better! Looking at this post reminds me of my previous roommate! He always kept preaching about this. I’ll forward this article to him. Fairly certain he’s going to have a great read. Many thanks for sharing! Hey very cool web site!! Man .. Beautiful .. Amazing .. I’ll bookmark your web site and take the feeds also…I am happy to find a lot of useful info here in the post, we need develop more techniques in this regard, thanks for sharing. . . . . .

Avatar_small
먹튀타운 said:
Tue, 20 Feb 2024 16:50:48 +0800

The post is really great. Information on various accessories organized in a basic and necessary way. Thank you for sharing this text. Matter is truly composed. This web writes my articles for me and often shares useful practical information. Please keep sharing many posts.

Avatar_small
먹튀검증사이트 said:
Tue, 20 Feb 2024 16:51:13 +0800

Hi, This is a nice article you shared great information I have read it thanks for giving such a wonderful Blog for the reader. Interesting post. I Have Been wondering about this issue. so thanks for posting. Pretty cool post.It 's really very nice and Useful post.Thanks. Your content is nothing short of brilliant in many ways. I think this is engaging and eye-opening material. Thank you so much for caring about your content and your readers.

Avatar_small
메이저사이트 said:
Tue, 20 Feb 2024 17:04:50 +0800

I and also my friends ended up following the nice thoughts from the blog and so quickly I got a terrible feeling I never expressed respect to the web site owner for those strategies. The women were certainly warmed to read through them and now have pretty much been making the most of them. Appreciate your turning out to be well thoughtful and also for selecting certain important issues most people are really needing to be aware of. Our own honest regret for not expressing appreciation to earlier. Needed to compose you that very little observation so as to say thanks a lot once again about the pleasing methods you have shown in this case. It was so surprisingly generous with you to offer unhampered precisely what some people would’ve sold as an electronic book to earn some profit on their own, precisely given that you could possibly have done it in case you desired. These techniques as well served like the fantastic way to realize that the rest have the identical zeal really like mine to understand a lot more pertaining to this problem. I am certain there are lots of more enjoyable sessions up front for individuals that scan your blog post.

Avatar_small
안전놀이터 said:
Tue, 20 Feb 2024 17:05:08 +0800

Thank you for taking the time to discuss this. I'd like to learn more about this topic. If possible, could you please update the blog with additional information once you gain expertise? It's very helpful to me.

Avatar_small
토토탐험대 said:
Tue, 20 Feb 2024 17:05:51 +0800

Thank you for taking the time to discuss this. I'd like to learn more about this topic. If possible, could you please update the blog with additional information once you gain expertise? It's very helpful to me.

Avatar_small
온라인카지노순위 said:
Tue, 20 Feb 2024 18:01:30 +0800

Are you curious about how to unlock your child's full potential? Look no further than their plate! It may come as a surprise, but the food your children eat plays a crucial role in their brain development and intelligence.

Avatar_small
카지노사이트 said:
Tue, 20 Feb 2024 18:01:50 +0800

The post is really great. Information on various accessories organized in a basic and necessary way. Thank you for sharing this text. Matter is truly composed. This web writes my articles for me and often shares useful practical information. Please keep sharing many posts.

Avatar_small
토토사이트추천 said:
Tue, 20 Feb 2024 18:02:09 +0800

Thank you for taking the time to discuss this. I'd like to learn more about this topic. If possible, could you please update the blog with additional information once you gain expertise? It's very helpful to me.

Avatar_small
เว็บไซท์ แทงบอลออนไล said:
Tue, 20 Feb 2024 18:33:01 +0800

Hey there, I think your blog might be having browser compatibility issues. When I look at your blog in Safari, it looks fine but when opening in Internet Explorer, it has some overlapping. I just wanted to give you a quick heads up! Other then that, awesome blog!

Avatar_small
메이저사이트 said:
Tue, 20 Feb 2024 18:33:24 +0800

Hey there, I think your blog might be having browser compatibility issues. When I look at your blog in Safari, it looks fine but when opening in Internet Explorer, it has some overlapping. I just wanted to give you a quick heads up! Other then that, awesome blog!

Avatar_small
사설토토사이트 said:
Tue, 20 Feb 2024 18:34:02 +0800

Well said. I in no way concept I ought to accept as true with this opinion, but I’m beginning to view topics from a specific view. I actually have to research extra on this as it seems very thrilling. One hassle I don’t understand despite the fact that is how the whole lot is associated collectively.

Avatar_small
먹튀신고 said:
Tue, 20 Feb 2024 18:42:37 +0800

This is a truly good site post. Not too many people would actually, the way you just did. I am really impressed that there is so much information about this subject that have been uncovered and you’ve done your best, with so much class. If wanted to know more about green smoke reviews, than by all means come in and check our stuff.

Avatar_small
토토사이트 said:
Tue, 20 Feb 2024 18:42:57 +0800

Excellent blog! Do you have any recommendations for aspiring writers? I’m planning to start my own blog soon but I’m a little lost on everything. Would you recommend starting with a free platform like WordPress or go for a paid option? There are so many options out there that I’m totally confused .. Any ideas? Thanks!

Avatar_small
메이저사이트 said:
Tue, 20 Feb 2024 20:30:26 +0800

You really make it seem so easy with your presentation but I find this topic to be really something that I think I would never understand. It seems too complex and extremely broad for me. I’m looking forward for your next post, I’ll try to get the hang of it!

Avatar_small
토토사이트 said:
Tue, 20 Feb 2024 20:30:48 +0800

Hey there, I think your blog might be having browser compatibility issues. When I look at your blog in Safari, it looks fine but when opening in Internet Explorer, it has some overlapping. I just wanted to give you a quick heads up! Other then that, awesome blog!


Login *


loading captcha image...
(type the code from the image)
or Ctrl+Enter