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A subring of a matrix ring is again a matrix ring. Over a rng, one can form matrix rngs. When R is a commutative ring, the matrix ring M n (R) is an associative algebra over R, and may be called a matrix algebra. In this setting, if M is a matrix and r is in R, then the matrix rM is the matrix M with each of its entries multiplied by r.
Rings are a more general notion than fields in that a division operation need not exist. The very same addition and multiplication operations of matrices extend to this setting, too. The set M(n, R) (also denoted M n (R) [7]) of all square n-by-n matrices over R is a ring called matrix ring, isomorphic to the endomorphism ring of the left R ...
The topology of SL(n, R) is the product of the topology of SO(n) and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of (n + 2)(n − 1)/2-dimensional Euclidean ...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
The general linear group GL(n, R) over the field of real numbers is a real Lie group of dimension n 2. To see this, note that the set of all n×n real matrices, M n (R), forms a real vector space of dimension n 2. The subset GL(n, R) consists of those matrices whose determinant is non-zero.
It is the group of complex orthogonal matrices, complex matrices whose product with their transpose is the identity matrix. As in the real case, O( n , C ) has two connected components. The component of the identity consists of all matrices of determinant 1 in O( n , C ) ; it is denoted SO( n , C ) .
The group SL(2, R) acts on its Lie algebra sl(2, R) by conjugation (remember that the Lie algebra elements are also 2 × 2 matrices), yielding a faithful 3-dimensional linear representation of PSL(2, R). This can alternatively be described as the action of PSL(2, R) on the space of quadratic forms on R 2. The result is the following representation:
Let A be an m × n matrix, with row vectors r 1, r 2, ..., r m. A linear combination of these vectors is any vector of the form + + +, where c 1, c 2, ..., c m are scalars. The set of all possible linear combinations of r 1, ..., r m is called the row space of A.