Direct Sum of Matrices

1. Direct Sum of Matrices can be calculated for Any 2 Matrices of Any Dimesions.
2. Given a \(M \times N\) Matrix \(A\) having elements \(a_{ij}\) and a \(P \times Q\) Matrix \(B\) having elements \(b_{ij}\), the Direct Sum of Matrices \(A\) and \(B\) is calculated as given below
   
   \(A \boxplus B = \begin{bmatrix}\textbf{A} & 0\\0 & \textbf{B}\end{bmatrix}= \begin{bmatrix} a_{11} & \cdots & a_{1n} & 0 & \cdots & 0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & \cdots & a_{mn} & 0 & \cdots & 0 \\ 0 & \cdots & 0 & b_{11} & \cdots & b_{1q} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & b_{p1} & \cdots & b_{pq} \end{bmatrix} \)
   
   The Resultant Matrix has \(M+P\) Number of Rows and \(N+Q\) Number of Columns.
3. Following example calculates the Direct Sum of Matrices \(A\) and \(B\) as given below
   
   \(A=\begin{bmatrix}7 & -2 & 3 \\4 & -7 & 6\end{bmatrix}\hspace{.5cm}B=\begin{bmatrix}2 & 9 \\ 11 & 3\end{bmatrix}\)
   
   \( A \boxplus B = \begin{bmatrix} 7 & -2 & 3 \\ 4 & -7 & 6 \end{bmatrix} \boxplus \begin{bmatrix}2 & 9 \\ 11 & 3\end{bmatrix} = \begin{bmatrix} 7 & -2 & 3 & 0 & 0 \\ 4 & -7 & 6 & 0 & 0 \\ 0 & 0 & 0 & 2 & 9 \\ 0 & 0 & 0 & 11 & 3 \end{bmatrix}\)