Let $A \in \mathbb{C}^{N \times M}$ and $D \in \mathbb{C}^{M \times M}$ each be a D999: Complex matrix such that

(i)	\begin{equation} A = \begin{bmatrix} A_{1, 1} & A_{1, 2} & \cdots & A_{1, M} \\ A_{2, 1} & A_{2, 2} & \vdots & A_{2, M} \\ \vdots & \cdots & \ddots & \vdots \\ A_{N, 1} & A_{N, 2} & \cdots & A_{N, M} \end{bmatrix} \end{equation}
(ii)	\begin{equation} D = \begin{bmatrix} D_{1, 1} & 0 & \cdots & 0 \\ 0 & D_{2, 2} & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & D_{M, M} \end{bmatrix} \end{equation}

Then \begin{equation} A D = \begin{bmatrix} A_{1, 1} D_{1, 1} & A_{1, 2} D_{2, 2} & \cdots & A_{1, M} D_{M, M} \\ A_{2, 1} D_{1, 1} & A_{2, 2} D_{2, 2} & \vdots & A_{2, M} D_{M, M} \\ \vdots & \cdots & \ddots & \vdots \\ A_{N, 1} D_{1, 1} & A_{N, 2} D_{2, 2} & \cdots & A_{N, M} D_{M, M} \end{bmatrix} \end{equation}

Let $A \in \mathbb{C}^{N \times M}$ and $D \in \mathbb{C}^{M \times M}$ each be a D999: Complex matrix such that

(i)	$A_1, A_2, \, \ldots, \, A_M \in \mathbb{C}^{N \times 1}$ are each a D5689: Complex column matrix
(ii)	\begin{equation} A = \begin{bmatrix} A_1 & A_2 & \cdots & A_M \end{bmatrix} \end{equation}
(iii)	\begin{equation} D = \begin{bmatrix} D_1 & 0 & \cdots & 0 \\ 0 & D_2 & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & D_M \end{bmatrix} \end{equation}

Then \begin{equation} A D = \begin{bmatrix} A_1 D_1 & A_2 D_2 & \cdots & A_M D_M \end{bmatrix} \end{equation}

Let $A \in \mathbb{C}^{N \times M}$ and $D \in \mathbb{C}^{M \times M}$ each be a D999: Complex matrix such that

(i)	\begin{equation} A = \begin{bmatrix} A_{1, 1} & A_{1, 2} & \cdots & A_{1, M} \\ A_{2, 1} & A_{2, 2} & \vdots & A_{2, M} \\ \vdots & \cdots & \ddots & \vdots \\ A_{N, 1} & A_{N, 2} & \cdots & A_{N, M} \end{bmatrix} \end{equation}
(ii)	\begin{equation} \Lambda = \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \vdots & 0 \\ \vdots & \cdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_3 \end{bmatrix} \end{equation}

Then \begin{equation} A \Lambda = \begin{bmatrix} A_{1, 1} \lambda_1 & A_{1, 2} \lambda_2 & \cdots & A_{1, M} \lambda_M \\ A_{2, 1} \lambda_1 & A_{2, 2} \lambda_2 & \vdots & A_{2, M} \lambda_M \\ \vdots & \cdots & \ddots & \vdots \\ A_{N, 1} \lambda_1 & A_{N, 2} \lambda_2 & \cdots & A_{N, M} \lambda_M \end{bmatrix} \end{equation}