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

*To*: mathgroup at smc.vnet.net*Subject*: [mg125561] Re: Eigenvalues, eigenvectors, matrix ranks, determinants, and all that stuff*From*: Konstantin <kparchevsky at gmail.com>*Date*: Mon, 19 Mar 2012 04:54:49 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

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"?