Let us say that the optimization problem can be posed in the matrix form as given below
$\min_{\mathbf{X} \in \mathbb{R}^{N \times K}} \left\|\mathbf{Y}-\mathbf{X}\right\|_F^2$ s.t. $\mathbf{A}\mathbf{X} = \mathbf{B}$, where $\mathbf{Y} \in \mathbb{R}^{N \times K}$, $\mathbf{A} \in \mathbb{R}^{M \times N}$, and $\mathbf{B} \in \mathbb{R}^{M \times K}$.
Question:
Without vectorizing the formulation, can the Lagrangian be defined as
$L(\mathbf{X},\mathbf{\Lambda}) = \left\|\mathbf{Y}-\mathbf{X}\right\|_F^2 + {\rm trace}\left(\mathbf{\Lambda}^T \left(\mathbf{A}\mathbf{X} - \mathbf{B} \right) \right)$ ?
If not, then how to construct a Lagrangian in the matrix form? Thank you.
EDIT: See this How to set up Lagrangian optimization with matrix constrains .
