## Direct Sum of 2 Matrices

## Answers to Questions

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

The addition by **direct sum** does not require any calculation, copy 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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