I came across following derivation: $$A^{T}Ax-A^{T}b+2\lambda x=0$$ $$x=(A^{T}A+2\lambda I)^{-1}A^{T}b$$
where $\lambda\ \epsilon R$ and $I$ is an identity matrix(with the same size as $A^{T}A$).
My question is why we can ''transform'' scalar value $\lambda$ into scalar matrix $\lambda I$. What is the justification of scalar and matrix addition? I know that, according to any Algebra book, we are allowed to do such thing. But why?
Since $Ix=x$ for any vector $x$ (of the appropriate size), we also have $(\lambda I)x=\lambda x$. It wouldn't make sense to factor $A^TAx+2\lambda x$ as $(A^TA+2\lambda)x$ (how do you add a scalar to a matrix?) but it does make sense to rewrite $2\lambda x$ as $2\lambda I x$ and then factor.