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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-module R n. [58] If the ring R is commutative, that is, its multiplication is commutative, then the ring M(n, R) is also an associative algebra over R.
Let R be a local ring. There is a determinant map from the matrix ring GL(R ) to the abelianised unit group R × ab with the following properties: [1] The determinant is invariant under elementary row operations; The determinant of the identity matrix is 1; If a row is left multiplied by a in R × then the determinant is left multiplied by a
The matrix ring M n (R) can be identified with the ring of endomorphisms of the free right R-module of rank n; that is, M n (R) ≅ End R (R n). Matrix multiplication corresponds to composition of endomorphisms. The ring M n (D) over a division ring D is an Artinian simple ring, a special type of semisimple ring.
The dual of a right module is a left module, and vice versa. The transpose of a matrix must be viewed as a matrix over the opposite division ring D op in order for the rule (AB) T = B T A T to remain valid. Every module over a division ring is free; that is, it has a basis, and all bases of a module have the same number of elements.
In algebra, the Wedderburn–Artin theorem is a classification theorem for semisimple rings and semisimple algebras.The theorem states that an (Artinian) [a] semisimple ring R is isomorphic to a product of finitely many n i-by-n i matrix rings over division rings D i, for some integers n i, both of which are uniquely determined up to permutation of the index i.
If is a singular matrix of rank , then it admits an LU factorization if the first leading principal minors are nonzero, although the converse is not true. [8] If a square, invertible matrix has an LDU (factorization with all diagonal entries of L and U equal to 1), then the factorization is unique. [7]
Decomposition: = where C is an m-by-r full column rank matrix and F is an r-by-n full row rank matrix Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A , [ 2 ] which one can apply to obtain all solutions of the linear system A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } .
Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups.These are the cyclic groups of prime order.. If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal.