Let us say that the optimization problem can be posed in the matrix form as given below
$$\min_{\mathbf{X} \in \mathbb{R}^{N \times K}} \left\|\mathbf{Y}-\mathbf{X}\right\|_{{\rm F}}^2 \quad \text{subject to} \quad \mathbf{A}\mathbf{X} = \mathbf{B}$$
where $\mathbf{Y} \in \mathbb{R}^{N \times K}$, $\mathbf{A} \in \mathbb{R}^{M \times N}$, and $\mathbf{B} \in \mathbb{R}^{M \times K}$. Without vectorizing the formulation, can the Lagrangian be defined as follows?
$$L(\mathbf{X},\mathbf{\Lambda}) = \left\|\mathbf{Y}-\mathbf{X}\right\|_{{\rm F}}^2 + {\rm trace}\left(\mathbf{\Lambda}^T \left(\mathbf{A}\mathbf{X} - \mathbf{B} \right) \right)$$
If not, then how to construct a Lagrangian in the matrix form? Thank you.
