Transpose in the context of Special orthogonal group

In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function ⁠${\displaystyle f:A\rightarrow A}$⁠, where A is a set; the function ⁠${\displaystyle f}$⁠ is a unary operation on A.

Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and superscripts (e.g. transpose A). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument.

In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order ${\displaystyle n}$. Any two square matrices of the same order can be added and multiplied.

Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if ${\displaystyle R}$ is a square matrix representing a rotation (rotation matrix) and ${\displaystyle \mathbf {v} }$ is a column vector describing the position of a point in space, the product ${\displaystyle R\mathbf {v} }$ yields another column vector describing the position of that point after that rotation. If ${\displaystyle \mathbf {v} }$ is a row vector, the same transformation can be obtained using ${\displaystyle \mathbf {v} R^{\mathsf {T}}}$, where ${\displaystyle R^{\mathsf {T}}}$ is the transpose of ${\displaystyle R}$.

In mathematics, a linear algebraic group is a subgroup of the group of invertible ${\displaystyle n\times n}$ matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation ${\displaystyle M^{T}M=I_{n}}$ where ${\displaystyle M^{T}}$ is the transpose of ${\displaystyle M}$.

Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and Kolchin (1948). In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today.