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

Matrix Direct Sum - dCode

Tag(s) : Matrix

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

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.

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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