Triangular matrix ring information

In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule. If T {\displaystyle T} and U...

abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set...

hereditary. If S is a von Neumann regular ring with an ideal I that is not a direct summand, then the triangular matrix ring [ S / I S / I 0 S ] {\displaystyle...

square matrix over a commutative ring is invertible if and only if its determinant is a unit (i.e. multiplicatively invertible element) of that ring. The...

both upper- and lower-triangular. A diagonal matrix is symmetric. The identity matrix In and zero matrix are diagonal. A 1×1 matrix is always diagonal....

commutative ring and A is an n × n matrix with entries from R. The (i, j)-minor of A, denoted Mij, is the determinant of the (n − 1) × (n − 1) matrix that results...

a ring is right and left Noetherian then the left and right global dimensions and the weak global dimension are all the same. The triangular matrix ring...

or left Kasch. The two-by-two upper (or lower) triangular matrix ring is not right or left Kasch. A ring with right socle zero (i.e. s o c ( R R ) = {...

element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix M is a unipotent...

algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It...

square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the...

of these vectors. A matrix whose elements above the main diagonal are all zero is called a lower triangular matrix, while a matrix whose elements below...

a ring is right and left Noetherian then the left and right global dimensions and the weak global dimension are all the same. The triangular matrix ring...

mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities are the...

of noncommutative rings: The matrix ring of n-by-n matrices over the real numbers, where n > 1 Hamilton's quaternions Any group ring constructed from a...

generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence...

j. Now define the N × N table (square matrix) whose (i, j)th entry is m(Gi, Gj). This matrix is lower triangular, and the elements on the diagonal are...

blockwise matrix inversion. While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. The Woodbury matrix identity...

a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional...

triangular matrix with zeros along the main diagonal is nilpotent, with index ≤ n {\displaystyle \leq n} [citation needed]. For example, the matrix B...

many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix L with all diagonal...

are both upper triangular. In other words, if A 1 , … , A k {\displaystyle A_{1},\ldots ,A_{k}} commute, there exists a similarity matrix P {\displaystyle...

take R to be the ring of all upper triangular matrices. However, it is not always the case that the injective hull of a ring has a ring structure, as an...

H if there is a square unimodular matrix U where H=AU and H has the following restrictions: H is lower triangular, hij = 0 for i < j, and any columns...

non-commutative finite ring with 1 has the order of a prime cubed, then the ring is isomorphic to the upper triangular 2 × 2 matrix ring over the Galois field...