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Mathematica problem, nontrivial solution for matrix equation Ax=0 
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#1
Sep711, 01:34 PM

P: 15

Hey, how can i get a non trivial solution from matrix equation Ax=0
more precisely, i want to calculate eigenvectors : (M a_1*I)x = 0, i keep getting x=0. 


#2
Sep711, 03:57 PM

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P: 17,306

Are you sure there is a nontrivial solution?
PS you may want to post here: http://www.physicsforums.com/forumdisplay.php?f=189 


#3
Sep911, 09:29 AM

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Thanks
PF Gold
P: 39,552

The equation Ax= 0 has a nontrivial solution if and only if A is not onetoone. That is the same as saying that its determant is 0 and that it has 0 as an eigenvalue. The standard way to find an eigenvalue, [itex]\lambda[/itex] for matrix A is to solve the equation [itex]det(A \lambda I)= 0[/itex]. If A is an n by n matrix, that will be a polynomial equation of degree n and so has n solutions (not necessarily all distinct, not necessarily real).
IF [itex]\lambda[/itex] really is an eigenvalue, then [itex]Ax= \lambda x[/itex] or [itex] Ax \lambda x= (A \lambda I)x= 0[/itex] has, by definition of "eigenvalue", a nontrivial solution. That is, some of the equations you get by looking at individual components will be dependent. Note that x= 0 always will be a solution, just not the only one. Perhaps if you posted a specific example, we could point out errors. The most obvious one, if you "keep getting x=0", is that what you think is an eigenvalue really isn't! 


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