norm(x) is a convex function, so a constraint norm(x) <= real constant
is convex and follows CVX 'x DCP rules.

norm(x) >= real constant
and norm(x) == real constant
are not convex and are not allowed n CVX.

So your problem as described is non-convex.

However, because you have specified an objective minimize(0)
you just have a feasibility problem, which means you could try to use a convex objective to help you out. I played around for a minute with different objective functions, and discovered that minimize(x(2)) or maximize(x(2)) produce a solution for which norm(x) = 2, and therefore satisfy your original feasibility problem.

cvx_begin sdp
variables x(2)
maximize(x(2))
subject to
[eye(2) A*x;(A*x)' 7]>=0;
norm(x,2)<=2;
cvx_end
Calling SeDuMi 1.34: 10 variables, 3 equality constraints
For improved efficiency, SeDuMi is solving the dual problem.
------------------------------------------------------------

Edited out solver output
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.72721

Changing the objective function to minimize(x(2)) results in the optimal x being the negative of that above, and so is another feasible solution. I don’t claim that these are the only feasible solutions, but given that you specified a feasibility problem, even if it had been convex and allowed by CVX, CVX would have only provided you a single feasible solution.

The problem is non-convex. I don’t see how to solve it in a general way with CVX.

For your first version, maybe I was just lucky to stumble upon a solution, or as I was thinking, maybe the RHS value of 2 was not "random’, and so something was going on which made that work out. Specifically, the optimal x was was just on the boundary of feasibility of the LMI, i.e., minimum LHS eigenvalue = 0, and that corresponded to norm(x) = 2.

I’m certain this FAQ has reduced the number of questions like this on the forum. And still: 90% of the posts we mark as Nonconvex are fully addressed by it.