Existance of solutions to set of quadratic equations

[Leo321]

We have two unknown vectors x,y with real non negative values.
We can assume ||x||=||y||=1.
There are m known nxn matrices with real values.
We have m equations of the form:
y^TA₁x=0
...
y^TA_mx=0

What are the conditions for the existence of solutions for x,y?
What is the minimal number m and the condition on the matrices(linear independence, etc) for there to be no solutions?

Thanks

 

[Leo321]

Any ideas?