Matrix Subtraction

\( M_1=[a_{ij}] \) is a matrix of \( m \) lines and \( n \) columns (with \( m = n \) in the case of a square matrix) and \( M_2=[b_{ij}] \) is another matrix of \( m \) lines and \( n \) columns.

The subtraction of these 2 matrices \( M_1 - M_2 = [c_{ij}] \) is an unchanged size matrix with \( m \) lines and \( n \) columns, such as : $$ \forall i, j : c_{ij} = a_{ij}-b_{ij} $$

Subtracting matrices is only possible with 2 matrices of the same shape. The calculation consists in subtracting the elements in the same position in each matrix.

Example: $$ \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} = \begin{bmatrix} 7-1 & 8-2 \\ 9-3 & 10-4 \\ 11-5 & 12-6 \end{bmatrix} = \begin{bmatrix} 6 & 6 \\ 6 & 6 \\ 6 & 6 \end{bmatrix} $$

The subtraction operation (or addition) of matrices is only defined with identical shapes matrices. Another operation called direct sum allows the use of matrices of different sizes and can be generalized to subtraction.