Editing Square matrix (section)

{{Short description|Matrix with the same number of rows and columns}}
[[File:Arbitrary square matrix.gif|thumb|A square matrix of order 4. The entries <math>a_{ii}</math> form the [[main diagonal]] of a square matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements {{math|1=''a''<sub>11</sub> = 9}}, {{math|1=''a''<sub>22</sub> = 11}}, {{math|1=''a''<sub>33</sub> = 4}}, {{math|1=''a''<sub>44</sub> = 10}}.]]
In [[mathematics]], a '''square matrix''' is a [[Matrix (mathematics)|matrix]] with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order {{nowrap|<math>n</math>.}} Any two square matrices of the same order can be added and multiplied. 

Square matrices are often used to represent simple [[linear transformation]]s, such as [[Shear mapping|shearing]] or [[Rotation (mathematics)|rotation]]. For example, if <math>R</math> is a square matrix representing a rotation ([[rotation matrix]]) and <math>\mathbf{v}</math> is a [[column vector]] describing the [[Position (vector)|position]] of a point in space, the product <math>R\mathbf{v}</math> yields another column vector describing the position of that point after that rotation. If <math>\mathbf{v}</math> is a [[row vector]], the same transformation can be obtained using {{nowrap|<math>\mathbf{v} R^{\mathsf T}</math>,}} where <math>R^{\mathsf T}</math> is the [[transpose]] of {{nowrap|<math>R</math>.}}