# Can this matrix equation be solved for X?

$(A^TA+aI)^{-1}A^T = A^TX$

where A and X are real-matrices and a is a positive non-zero real number. A is non-zero and you may assume that A is such that there is a unique X that satisfies the above equation.

-

$$A^T=(A^TA+aI)A^TX$$ $$A^T=A^T(AA^T+aI)X$$
and now if $X=(AA^T+aI)^{-1}$ it is solved.
$(A^TA+aI)A^T=(A^TAA^T+aIA^T)=A^T(AA^T+aI)$ –  adam W Oct 4 '12 at 0:52