I am trying to implement an armijo backtracking algorithm for this function: $$ f(x) = \frac{1}{2} ||Ax - b||_2^2, \hspace{10px} \nabla f(x) = A^T(Ax - b) \hspace{120px}b,x\in R^{m\times n}$$
But the condition in this case compares a scalar value with a matrix. $$f(x_k+p_k)\le f(x_k)+c_1\alpha \nabla f(x_k)^T p_k\hspace{20px}p_k=-\nabla f(x_k)$$ How can I modify the armijo condition?