How to Derive this Matrix Equation - Converting Lagrangian in Proximal Operator Problem Form $$ \left< Z,X-L-S \right> \quad +\quad \frac { r }{ 2 } \left\| X-L-S \right\|_F^2 \quad =\quad \frac { r }{ 2 } { \left\| L-\left( X-S+\frac {Z}{r}  \right)  \right\|  }_F^2 $$
I think $ \left< Z,X-L-S \right>$ should be $\operatorname{Tr}(Z^T(X-L-S))$. This equation is presented in a paper, but the author doesn't show the derivation. See Robust Principal Component Analysis on Graphs at page 11.

Note that the left side of
$$ \left< Z,X-L-S \right> \quad +\quad \frac { r }{ 2 } \left\| X-L-S \right\|_F^2 \quad =\quad \frac { r }{ 2 } { \left\| L-\left( X-S+\frac {Z}{r}  \right)  \right\|  }_F^2 $$
can be written as
$$
\operatorname{tr}(Z^T(X-L-S)) +  \frac { r }{ 2 } \operatorname{tr}( (X-L-S)^T(X-L-S) ).$$
And this is the same as
$$
\operatorname{tr}(Z^T(X-L-S)) + \operatorname{tr}(\frac { r }{ 2 } (X-L-S)^T(X-L-S) ) = 
\operatorname{tr}\left(Z^T(X-L-S)) + \frac { r }{ 2 } (X-L-S)^T(X-L-S) )\right)
= \operatorname{tr}\left(\left((Z^T) + \frac { r }{ 2 } (X-L-S)^T\right)(X-L-S) )\right)
= \frac { r }{ 2 }\operatorname{tr}\left(\left(\frac{2 Z^T}{r} + (X-L-S)^T\right)(X-L-S) )\right)
= \frac { r }{ 2 }\operatorname{tr}\left(\left(\frac{2 Z}{r} + X-L-S\right)^T(X-L-S) )\right).
$$
The right hand side on the other hand can be written as 
$$\frac { r }{ 2 } { \left\| L-\left( X-S+\frac {Z}{r}  \right)  \right\|  }_F^2 = 
\frac { r }{ 2 } \operatorname{tr} \left( \left(L-X+S-\frac {Z}{r}  \right)^T \left(L-X+S-\frac{Z}{r} \right)  \right) 
=
\frac { r }{ 2 } \operatorname{tr} \left( \left(L-X+S-\frac {2Z}{r}+\frac {Z}{r}  \right)^T \left(L-X+S-\frac{2Z}{r} +\frac{Z}{r} \right)  \right) 
=
\frac { r }{ 2 } \operatorname{tr} \left( \left(-L+X-S+\frac {2Z}{r}-\frac {Z}{r}  \right)^T \left(-L+X-S+\frac{2Z}{r} -\frac{Z}{r} \right)  \right)
=
\frac { r }{ 2 } \operatorname{tr} \left( \left((-L+X-S+\frac {2Z}{r})-\frac {Z}{r}  \right)^T \left((-L+X-S)+(\frac{2Z}{r} -\frac{Z}{r}) \right)  \right)
=
\frac { r }{ 2 } \operatorname{tr} \left((-L+X-S+\frac {2Z}{r})^T(-L+X-S)-\frac {Z^T}{r}(-L+X-S)   + (-L+X-S+\frac {2Z}{r})^T\frac{Z}{r}-\frac {Z^T}{r}\frac{Z}{r}  \right) 
=
\frac { r }{ 2 } \operatorname{tr} \left((-L+X-S+\frac {2Z}{r})^T(-L+X-S) + (\frac {2Z}{r})^T\frac{Z}{r}-\frac {Z^T}{r}\frac{Z}{r}  \right) 
$$
