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MathGroup Archive 2004

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Re: please solve [LinearAlgebra]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg53039] Re: please solve [LinearAlgebra]
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 21 Dec 2004 05:19:16 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <cprk0m$r87$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <cprk0m$r87$1 at smc.vnet.net>, rcmcll at yahoo.com wrote:

> Could someone please solve this symbolically?
> This is just the ols formula for beta-hat but I need a symbolic
> solution for this special case.
> 
> b = inv(x'x)x'y
> 
> where
> 
> x =  1  t  t^2
> 1  t  t^2
> 1  t  t^2
> 
> 
> and simplify simplify simplify!!

For a singular matrix you need to compute the PseudoInverse:

  X = Table[t^(i-1), {3}, {i, 3}]

  Y = Table[y[i], {i, 3}]

  b = PseudoInverse[X] . Y // Simplify

Assuming that t is real, you obtain

  Simplify[b, Element[t, Reals]]

  {    (y[1] + y[2] + y[3])/(3 (t^4 + t^2 + 1)), 
    t  (y[1] + y[2] + y[3])/(3 (t^4 + t^2 + 1)), 
   t^2 (y[1] + y[2] + y[3])/(3 (t^4 + t^2 + 1))}

You get exactly the same result if you compute

  PseudoInverse[Transpose[X] . X] . Transpose[X] . Y

and simplify.

In 5.1, if you paste the following cell into a Notebook and click on the 
button it will take you to the advanced documentation:

  Cell[TextData[{"See the ",  
   ButtonBox["Advanced Documentation",    
      ButtonData:>"Advanced Documentation: Linear Algebra",    
      ButtonStyle->"RefGuideLink"],  
   " for Linear Algebra for more information, particularly the section 
on pseudoinverse in the Basic Operations section of Section 4."}], 
   "Text"]

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 6488 2734
School of Physics, M013                         Fax: +61 8 6488 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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