Matrix and symmetry groups
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The idea of a matrix group
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Matrices are mathematical objects that are generally subject to two matrix binary operations (1) addition of matrices (2) multiplication of matrices. It is, therefore, possible to form groups of matrices whenever one or other of these binary operations is clearly defined, and whenever the resultant objects are of the same form. For example, the set of all matrices of the form [Equation goes here - download the original to see it.] where a, b are real numbers forms a matrix group under matrix addition. [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The set is also closed under the operation of addition of matrices, since [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Matrix addition is associative. This verifies that this is a group, and illustrates the formation of matrix groups. Matrices are intimately connected with transformations of the plane and consequently, corresponding to the transformations of the plane and plane figures there are matrix groups corresponding to the set of linear translations is the set of all matrices of the form [Equation goes here - download the original to see it.] where a, b ¡ under matrix addition. The set of all rotations about the origin and reflections in lines passing through the origin corresponds to a group of matrices called the general orthogonal group. To specify the form of matrices of this type we must first define the transpose of a matrix. The transpose of a matrix A is the matrix AT obtained when the rows of A are written as columns. Example [Equation goes here - download the original to see it.]
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Contents of
Matrix and symmetry groups
1 The idea of a matrix group
2 Some special matrix groups
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