🚀 go-pugleaf

Ok here it is......

First start with a system that is hyperbolic or at least quasi-hyperbolic.
Such as...


        [U]     +     [F]    =   [0]                         (1)
           t             x            


where [F] is (or has the property of being) homogeneous of degree one in [U]. 
This means(I think..) that the above equation can be written as :

        [U]      +     [A][U]    = 0                           (2) 
           t                 x 	
		

Where [A] is given as d[F]/d[U] = [A].    
For the above system to be purely hyperbolic the eigenvalues of [A] must all
be real and distinct. If the eigenvalues are repeated the system can be
"hyperbolic" if a set of eigenvectors are linearly independent.
Given this hyperbolic system of equations (or at least quasi-hyperbolic), I
would like to show the following:



        [R]   +  [A][R]   = 0                                  (3)
           t           x            

	

	where [R] is a square matrix whose columns consist of the right eigenvectors of
[A]. 
This type of equation comes about if one tries to manipulate the second 
equation into a form via the substitution: [R][w] = [U]
Which gives:.....

        ([R]    +   [A][R] ) [w]  + [R][w]   +  [A][R][w]   = [0]   (4)
            t             x               t              x       

	Then we get the result:


                         [w]      +     [R]^(-1)[A][R][w]    = [0]    (5)
                            t                            x
Or:
                      \
        [w]       + [ /\ ] [w]   = [0]   
           t                  x         
                  \   
           where [/\] is a diagonal matrix having eigenvalues on its diagonal. 


So the bottom line is can you show equation (3) equals zero.? 
An engineer's way around that is to say [R] is a constant :) but 
since this news group is a math mecca I was hoping to find a more sophiscated
way! 


	Well, thanks in advance!  
	P.S. I hope the tabs and spaces show up as I typed them....



------------------------------------------------------------------------------


	Jim Miller				jhmiller@iastate.edu
	Department of Aerospace                 phone:  515-294-9497
	      Engineering 
	Iowa State University
	(maybe a graduate someday)

	"Failing to plan is planning to fail" - John Atella
	"The roof! The roof! The roof is on fiyah!........"