How to Solve a Homogeneous System with Norm Constraint?

[mnb96]

Hello,
how would you solve an homogeneous system of the form $A\mathbf{x}=0$, with the constrain $<\mathbf{x},\mathbf{x}>=1$. The matrix A is symmetric, but I don't know if it matters.
There should be a method involving eigenvalues, but strangely enough, I can't find it in any book.
Thanks!

 

[Hurkyl]

Why not just do the most straightforward thing? First solve Ax=0. Then among its solutions, solve <x,x>=0.

 

[mnb96]

The problem is that Ax=0 has a trivial solution which is x=0, and I am not interested in that.

 

[Hurkyl]

If x=0 is entire solution set to Ax=0, then what does that say about the solution set to your original problem?

 

[mnb96]

actually I don't know if x=0 is the entire solution set for the system. Hopefully it is not. I am interested in figuring out whether there are other solutions or not. and eventually find them.

 

[Hurkyl]

I am interested in figuring out whether there are other solutions or not. and eventually find them.

Then do that -- use your linear algebra to find all of them.

Then once you know all of them, you can find which of them satisfy your constraint.