BranchesHistoryFPLHelpLogin
Welcome guest
You're not logged in.
263 users online, thereof 0 logged in

Definition: Matrix and Vector Addition

Let \(A,B\in M_{m\times n}(F)\) be two matrices over a given field. The matrix addition $”+”$ is defined by

$$
C:=A+B=\pmatrix{
\alpha_{11} & \alpha_{12} & \ldots & \alpha_{1n} \cr
\alpha_{21} & \alpha_{22} & \ldots & \alpha_{2n} \cr
\vdots & \vdots & \ddots & \vdots \cr
\alpha_{m1} & \alpha_{m2} & \ldots & \alpha_{mn} \cr
}+\pmatrix{
\beta_{11} & \beta_{12} & \ldots & \beta_{1n} \cr
\beta_{21} & \beta_{22} & \ldots & \beta_{2n} \cr
\vdots & \vdots & \ddots & \vdots \cr
\beta_{m1} & \beta_{m2} & \ldots & \beta_{mn} \cr
}=\pmatrix{
\alpha_{11}+\beta_{11} & \alpha_{12}+\beta_{12} & \ldots & \alpha_{1n}+\beta_{1n} \cr
\alpha_{21}+\beta_{21} & \alpha_{22}+\beta_{22} & \ldots & \alpha_{2n}+\beta_{2n} \cr
\vdots & \vdots & \ddots & \vdots \cr
\alpha_{m1}+\beta_{m1} & \alpha_{m2}+\beta_{m2} & \ldots & \alpha_{mn}+\beta_{mn} \cr
}.
$$

The resulting matrix $C\in M_{m\times n}(F)$ is called a matrix sum of the two matrices \(A\) and \(B\).

Vector addition is a special case of matrix addition for matrices of the size $M_{m\times 1}(F)$ for column vectors

$$
\pmatrix{
\alpha_{1} \cr
\alpha_{2} \cr
\vdots \cr
\alpha_{m} \cr
}+\pmatrix{
\beta_{1} \cr
\beta_{2} \cr
\vdots \cr
\beta_{m} \cr
}=\pmatrix{
\alpha_{1}+\beta_{1}\cr
\alpha_{2}+\beta_{2}\cr
\vdots \cr
\alpha_{m}+\beta_{m} \cr
}.
$$

or of the size $M_{1\times n}(F)$ for row vectors.

$$ \pmatrix{\alpha_{1},&\alpha_{2},&\ldots, &\alpha_{n}} + \pmatrix{\beta_{1},&\beta_{2},&\ldots, &\beta_{n}}=\pmatrix{\alpha_{1}+\beta_{1},&\alpha_{2}+\beta_{2},&\ldots,& \alpha_{n}+\beta_{n}}.$$

| | | | | created: 2020-07-04 09:30:18 | modified: 2020-11-29 07:39:49 | by: bookofproofs | references: [8684]

Edit or AddNotationAxiomatic Method

This work was contributed under CC BY-SA 4.0 by:

This work is a derivative of:

Bibliography (further reading)

[8684] Axler, Sheldon: “Linear Algebra Done Right”, Springer, 2015, 3rd