Definition:Matrix Entrywise Addition/General

From ProofWiki
Jump to navigation Jump to search


Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$ be matrices all of order of $m \times n$.

Then the matrix entrywise sum of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$ is written $\mathbf A_1 + \mathbf A_2 + \ldots + \mathbf A_k$, and is defined as follows:


$\forall i \in \closedint 1 m, j \in \closedint 1 n: K_{i j} = \paren {a_1}_{i j} + \paren {a_2}_{i j} + \cdots + \paren {a_k}_{i j}$

where $\paren {a_l}_{i j}$ is the element of $\mathbf A_l$ whose indices are $\tuple {i, j}$.

Thus $\mathbf K = \sqbrk k_{m n}$ is the $m \times n$ matrix whose entries are made by performing the adding corresponding entries of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_k$.

This can be expressed in summation form as:

$\ds K = \sum_{j \mathop = 1}^k \mathbf A_j$

Also see

  • Results about matrix entrywise addition can be found here.