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