$A$ is an $m \times n$ matrix, $x$ is a $n \times 1$ vector, $y$ is an $m \times 1$ vector. Is this solution true?
Solution:
Let
$$ f\left( x \right) = \left( Ax-y \right) ^T\left( Ax-y \right) , $$
we have
$$ \nabla \left[ f\left( x \right) \right] ^l=l\left[ f\left( x \right) \right] ^{l-1}\nabla f\left( x \right) =2l\left[ \left( Ax-y \right) ^T\left( Ax-y \right) \right] ^{l-1}A^T\left( Ax-y \right). $$