Linear and multilinear algebra; matrix theory Definition

Matrix addition

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Definition

Let $R$ be a ring and let $A = (a_{ij})$ and $B = (b_{ij})$ be $m \times n$ matrices over $R$ . The sum $A + B$ is the $m \times n$ matrix with entries $(A + B)_{ij} = a_{ij} + b_{ij}$ for $1 \le i \le m$ and $1 \le j \le n$ .

Remarks

The sum is defined only for two matrices of the same size. Matrix addition is associative and commutative, the zero matrix is its identity element, and $-A = (-a_{ij})$ is the additive inverse of $A$ . Thus $M_{m \times n}(R)$ is an abelian group under addition. For example, $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 4 & 4 \end{pmatrix}$ .

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Matrix definition
Matrix addition definition

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Tag	`0BVXADM10P`
URI	`m2://0BVXADM10P`
Kind	definition
Writing	W1 stated
MSC	15A99
Defines	Matrix addition
Uses	Matrix

Matrix addition

Definition

Remarks

Reading path

Where this leads

Further reading

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