> [!NOTE] Definition (Matrix) > Let $S$ be a [[Sets|set]]. Let $m,n\in \mathbb{N}^{+}.$ An $m\times n$ matrix over $S$ is a [[Function|function]] from the [[Cartesian Product|cartesian product]] of two [[Integer Interval|integer intervals]] into $S$: $A: [1,\dots,m]\times[1,\dots,n]\to S.$ **Notation**: $A(i,j)$ is referred to as the $(i,j)$ entry $A,$ and commonly denoted by $a_{i,j}$ or $a_{ij}.$ A matrix is usually written as a rectangular array consisting of $mn$ numbers arranged in $m$ [[Matrix Row|rows]] and $n$ [[Matrix Column|columns]] where $a_{ij}$ is the entry in the $i$-th row and $j$-th column: $A=\begin{pmatrix} a_{11} & a_{12} & a_{13} &\dots &a_{1n} \\ a_{21} & a_{22} & a_{23} &\dots &a_{2n} \\ \vdots & \vdots & \vdots &\dots &\vdots \\ a_{m1} & a_{m2} & a_{m3} &\dots &a_{mn} \end{pmatrix}$This may be abbreviated by writing $A=(a_{ij})_{1\leq i,j\leq n}.$ # Properties The [[Dual of linear map|transpose]] of $A=(a_{ij})$ is given by $A^{T} = (a_{ji})$. **Algebra**: we define the following operations on $\text{Mat}_{mn}$ [[Matrix Scalar Multiplication]] & [[Matrix Product]]. Any $m\times n$ matrix determines a [[Left Multiplication Linear Map of Real Matrix|left multiplication linear map]], $\begin{align} L_{A}:\; &\mathbb{R}^{n\times l} \to \mathbb{R}^{m\times l} \\ &\underline{v} \mapsto A \underline{v} \end{align}$(note that linearity of $T$ follows from the distributivity of matrix multiplication). An $n\times n$ square matrix represent a [[Linear maps|linear operator]] (a linear from $\mathbb{R}^{n}$ to itself). An [[Eigenpair|eigenvector]] of a square matrix is one whose direction is unchanged after left multiplication by the matrix. The scaling factor is known as its corresponding *eigenvalue.*