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.