Re: Eigenvalues, eigenvectors, matrix ranks, determinants, and all that stuff

*To*: mathgroup at smc.vnet.net*Subject*: [mg125606] Re: Eigenvalues, eigenvectors, matrix ranks, determinants, and all that stuff*From*: Peter Pein <petsie at dordos.net>*Date*: Wed, 21 Mar 2012 05:46:09 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

Am 19.03.2012 10:56, schrieb Konstantin: > On Mar 16, 4:33 am, Bill Rowe<readn... at sbcglobal.net> wrote: >> On 3/15/12 at 12:31 AM, kparchev... at gmail.com (Konstantin) wrote: >> >> <snip> >> >>> Very often algebraic manipulations in Mathematica do not produce >>> expected result just because we and the program have different >>> defaul assumptions about type of variables. So I tried to specify >>> EVERYTHING about my variables explicitly ( @ means "belongs to", != >>> means "not equal" ) >>> $Assumptions = u@Reals&& u!=0&& Bx@Reals&& By@Reals&& Bz@Reals >>> && Bx!=0&& By!=0&& Bz!=0&& rho>0&& cs>0; >> >> Unless you have re-defined @ (an ill advised thing to do) you >> are not repeat not free to use it to do something other than the >> built-in definition. That is Mathematica interprets u@Reals as >> the function u with argument Reals, i.e: >> >> In[9]:= u@Reals >> >> Out[9]= u(\[DoubleStruckCapitalR]) >> >> If you want to declare u to be a real variable use the function >> Element that performs exactly this function That is the statement >> >> {u, Bx, By, Bz} \[Element] Reals >> >> defines the list of variables all to be real. >> >> And note, >> >> In[10]:= Reduce[{Element[x, Reals]&& x != 0}, x] >> >> Out[10]= x<0\[Or]x>0 >> >> That is stating x != 0 is sufficient to declare x to be real >> since the comparison of x to 0 is not valid unless x is real > > I do not use symbol @ in my code. I use standard symbol "belongs to" > to specify, that u is real. There is no equivalent for this symbol in > ASCII table. And I mentioned in brackets, that in my notation @ means > "belongs to". So, syntax is fine. What can you say about the main > point? Why does Mathematica calculates rank of matrix incorrectly and > how can I calculate my eigenvectors other than do simplifications > "manually"? > As you want Abs[Bxyz], put it into the matrix: A1 = {{u, rho, 0, 0, 0, 0, 0, 0}, {0, u, 0, 0, rho^(-1), 0, Abs[By]/(4*Pi*rho), Abs[Bz]/(4*Pi*rho)}, {0, 0, u, 0, 0, 0, -Abs[Bx]/(4*Pi*rho), 0}, {0, 0, 0, u, 0, 0, 0, -Abs[Bx]/(4*Pi*rho)}, {0, cs^2*rho, 0, 0, u, 0, 0, 0}, {0, 0, 0, 0, 0, u, 0, 0}, {0, Abs[By], -Abs[Bx], 0, 0, 0, u, 0}, {0, Abs[Bz], 0, -Abs[Bx], 0, 0, 0, u}}; or use oldA1 /.{pi->Pi,(b:Bx|By|Bz):>Abs[b]} the fourth eigenvalue/vector seems to be what you want: {val,vec}=Simplify[Eigensystem@A1][[All,4]] gives: {u+Abs[Bx]/(2 Sqrt[\[Pi]] Sqrt[rho]), {0,0,Abs[Bz]/(2 Sqrt[\[Pi]] Sqrt[rho] Abs[By]), -(1/(2 Sqrt[\[Pi]] Sqrt[rho])),0,0,-(Abs[Bz]/Abs[By]),1} } check: A1.vec-val vec//Simplify {0,0,0,0,0,0,0,0} further I get: In[9]:= MatrixRank[A1-val IdentityMatrix[8]] Out[9]= 7 and In[10]:= Solve[(A1-val IdentityMatrix[8]).Array[v,8]==0,Array[v,8]] Solve::vars: During evaluation of In[10]:= Solve::svars: Equations may not give solutions for all "solve" variables. >> Out[10]= {{v[1]->0,v[2]->0,v[4]->-((Abs[By] v[3])/Abs[Bz]),v[5]->0,v[6]->0,v[7]->-2 Sqrt[\[Pi]] Sqrt[rho] v[3],v[8]->(2 Sqrt[\[Pi]] Sqrt[rho] Abs[By] v[3])/Abs[Bz]}} to get the same vector as via Eigensystem, add v[8]==1 to the equations: In[11]:= Solve[(A1-val IdentityMatrix[8]).Array[v,8]==0&&v[8]==1,Array[v,8]] Out[11]= {{v[1]->0,v[2]->0,v[3]->Abs[Bz]/(2 Sqrt[\[Pi]] Sqrt[rho] Abs[By]),v[4]->-(1/(2 Sqrt[\[Pi]] Sqrt[rho])),v[5]->0,v[6]->0,v[7]->-(Abs[Bz]/Abs[By]),v[8]->1}} hth, Peter