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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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