In my research of linear algebra and optimization, I wish to modify the following well-known problem:



> 
> $ \min \lVert x-Ax \rVert$ subject to $ rank(A)\leq k $ where $ x $ is a given column vector and we optimize over matrices of bounded rank.
> 
> 
> 


My question is, can there be a way to solve the modified problem where the constraint is put on the norm (any matrix norm) of the matrix rather than its rank? I am not able to see a way to do this but perhaps it can be done or approximated. I thank all helpers.



Modification of a known optimization problem
This can be expressed (converted into) as a convex Second Order Cone Problem or SDP, depending on the norms used. CVX or another modeling tool can be used to convert the entered problem into a standard form for a solver. For example,



```
cvx_begin
variable A(m,n)
minimize(norm(x-A*x,p1))
norm(A,p2) <= bound_on_norm
cvx_end

```

p1 can be any number >= 1 or Inf (specifies p1-norm). p2 can be any of 1, 2, Inf, or 'fro' (Frobenius) to specify which norm to use. Or use norm\_nuc instead of norm in the constraint to use nuclear norm.


