Definition

A matrix is a grid of numbers organized in rows and columns, a rectangular array with $m$ rows and $n$ columns called an $m \times n$ matrix, like a spreadsheet of numbers packed into one mathematical object; the entry in row $i$, column $j$ is denoted $a_{ij}$. Matrices can be added (if the same dimensions), multiplied by scalars, and multiplied together (if dimensions are compatible), representing linear transformations, so the product $AB$ represents the composition of transformation $B$ first, then $A$, which is why matrix multiplication is not commutative. Formally, an $m \times n$ matrix over a field $F$ is an element of the vector space $M_{m,n}(F)$ of dimension $mn$, and the set $M_n(F)$ of square matrices forms a (non-commutative, for $n \ge 2$) ring; every linear map $T: F^n \to F^m$ corresponds to an $m \times n$ matrix relative to chosen bases, and Jordan normal form shows that every complex square matrix is similar to a canonical "almost diagonal" form encoding the multiplicities of its eigenvalues.