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...

in matrix theory. Square matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring. The...

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...

determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a 2 × 2 matrix is | a b c d | = a d − b c ,...

an invertible matrix is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can...

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....

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...

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

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...

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...

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...

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^{+}} ⁠ of a matrix ⁠ A {\displaystyle A} ⁠, often called the pseudoinverse, is the most widely known generalization of the inverse matrix. It was independently...

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...

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...

upper triangular matrix (that has more than one block of positive size). (Replacing non-zero entries in the matrix by one, and viewing the matrix as the...

from the fact that A is always similar to its Jordan form, an upper triangular matrix having λ1, ..., λn on the main diagonal. In contrast, the determinant...

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...

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...

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...

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...

right or left Kasch. The 2-by-2 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 ) = {...

Hy = b has an integer solution is easier than Ax = b because the matrix H is triangular.: 55  Many mathematical software packages can compute the Hermite...

square matrix, then the product of A with itself n times is called the matrix power. Also A 0 {\displaystyle A^{0}} is defined to be the identity matrix, and...

tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices...

rings. More exotic examples include the upper triangular matrices over a division ring Tn D, and the group ring F [ G ] {\displaystyle \mathbb {F} [G]} for...

square matrix of size n. Then: Cn (A) = det A. If A has one of the following properties, then so does Cr (A): Upper triangular, Lower triangular, Diagonal...