9.4 Matrix-Matrix Addition

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 9.4.1 Identity as Matrix Matrix Addition Example
Constructions: 9.5 Scalar-Matrix Multiplication
Generalizations: N/A

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Matrix-Matrix Addition

If $A$ and $B \in R^{m \times n}$ are matrices with matching dimensions (same number of rows and columns), then the matrix sum $A + B$ is the matrix obtained by adding the corresponding entries of $A$ and $B$ .

{matrix sum}_{i k} = a_{i k} + b_{i k}

An example of matrix-matrix addition with values in action:

[\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & 0 \end{matrix}] + [\begin{matrix} 2 & 2 & 0 \\ 0 & 0 & 1 \\ 9 & 4 & 2 \end{matrix}] = [\begin{matrix} 3 & 2 & 0 \\ 0 & 2 & 1 \\ 12 & 4 & 2 \end{matrix}]

A more general example of adding matrices $A + B$ would look like:

\begin{aligned} A + B & = [\begin{array}{c} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}] + [\begin{array}{c} b_{11} & b_{12} & \dots & b_{1 n} \\ b_{21} & b_{22} & \dots & b_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ b_{m 1} & b_{m 2} & \dots & b_{m n} \end{array}] \\ = [\begin{array}{c} a_{11} + b_{11} & a_{12} + b_{12} & \dots & a_{1 n} + b_{1 n} \\ a_{21} + b_{21} & a_{22} + b_{22} & \dots & a_{2 n} + b_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} + b_{m 1} & a_{m 2} + b_{m 2} & \dots & a_{m n} + b_{m n} \end{array}] \end{aligned}