Matrix Subtraction

Take 2 matrices of identical size: $ M_1=[a_{ij}] $ a matrix of $ m $ rows and $ n $ columns (with $ m = n $ in the case of a square matrix) and $ M_2=[b_{ij}] $ 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 defined with 2 matrices of the same shape (square 2x2, 3x3 or rectangular 2x3, 3x2, etc.). 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 marix subtraction operation is only defined with identical shapes matrices (as the operation of matrix addition). Another operation called direct sum allows the use of matrices of different sizes and can be generalized to subtraction.