S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
It is currently Wed, 19 Jun 2013 02:47:18 UTC

All times are UTC [ DST ]




Post new topic Reply to topic  [ 18 posts ]  Go to page Previous  1, 2
Author Message
 Post subject: Re: Matrix proof help PLZ!
PostPosted: Sat, 1 Oct 2011 10:39:00 UTC 
Offline
Moderator
User avatar

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6066
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
twinwings wrote:
I think...dimensions of Px=nxn and Px=I (I still have some trouble with this)


No! P_X need not be the identity matrix.

We know X^TX is invertible k-by-k, hence \mathop{\mathrm{rank}}X\geq k, hence n\geq k. If X=\begin{pmatrix}I_{k\times k}\\0_{(n-k)\times k}\end{pmatrix}, then X^TX=I_{k\times k}, so P_X=X(X^TX)^{-1}X^T=XX^T=\begin{pmatrix}I_{k\times k} & 0_{k\times(n-k)}\\0_{(n-k)\times k} & 0_{(n-k)\times(n-k)}\end{pmatrix}.

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


Top
 Profile  
 
 Post subject: Re: Matrix proof help PLZ!
PostPosted: Mon, 3 Oct 2011 03:55:50 UTC 
Offline
S.O.S. Newbie

Joined: Mon, 3 Oct 2011 03:54:01 UTC
Posts: 1
Hey guys, i was just wondering if anyone could shed some light on part (c)? I understand the previous 2 parts, but part (c) has me really lost for ideas. Thanks a ton.


Top
 Profile  
 
 Post subject: Re: Matrix proof help PLZ!
PostPosted: Mon, 3 Oct 2011 10:03:38 UTC 
Offline
Moderator
User avatar

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6066
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
chrisdelp wrote:
Hey guys, i was just wondering if anyone could shed some light on part (c)? I understand the previous 2 parts, but part (c) has me really lost for ideas. Thanks a ton.


Just calculate P_Y=Y(Y^TY)^{-1}Y^T when Y=XA.

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 18 posts ]  Go to page Previous  1, 2

All times are UTC [ DST ]


Who is online

Users browsing this forum: No registered users


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum

Search for:
Jump to:  
Contact Us | S.O.S. Mathematics Homepage
Privacy Statement | Search the "old" CyberBoard

users online during the last hour
Powered by phpBB © 2001, 2005-2011 phpBB Group.
Copyright © 1999-2013 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA