# Question on homogeneous systems

1. Oct 10, 2009

### 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!

2. Oct 10, 2009

### Hurkyl

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

3. Oct 10, 2009

### mnb96

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

4. Oct 10, 2009

### Hurkyl

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

5. Oct 10, 2009

### mnb96

actually I dont 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.

6. Oct 10, 2009

### Hurkyl

Staff Emeritus
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.