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Matrix Direct Sum

Tool to calculate the direct sums of matrices (formal computation). The matrix direct sum calculates the sum of N matrices that can be of different sizes.

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Matrix Direct Sum -

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# Matrix Direct Sum

## Direct Sum of 2 Matrices

### How to add 2 matrices with direct sum?

Given $M_1=[a_{ij}]$ a matrix of $m$ lines and $n$ columns and $M_2=[b_{ij}]$ a matrix of $p$ lines and $q$ columns (2x2, 2x3, 3x2, 3x3, etc).

The direct sum of these 2 matrices is noted with the character (circled plus sign) $M_1 \oplus M_2$ and is a matrix of $m+p$ lines and $n+q$ columns.

$$A \oplus B = \begin{bmatrix} [a_{ij}] & [0] \\ [0] & [b_{ij}] \end{bmatrix}$$

Example: $$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \oplus \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 & 0 & 0 \\ 4 & 5 & 6 & 0 & 0 \\ 0 & 0 & 0 & 7 & 8 \\ 0 & 0 & 0 & 9 & 10 \end{bmatrix}$$

Practically, the matrix addition by direct sum does not require any calculation, it only consist in copying the matrices diagonally, into a larger one, and fill with zeros.

The direct sum operation must be distinguished from the conventional operation of matrix addition, although it may take different size matrices, the result is not at all identical.

### How to add N matrices with direct sum?

The direct addition is generalizable to N matrices, but the order matter.

$$A \oplus B \oplus C = ( A \oplus B ) \oplus C \neq A \oplus ( B \oplus C )$$

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