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The inverse of an upper triangular matrix, if it exists, is upper triangular. The product of an upper triangular matrix and a scalar is upper triangular. Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size.
A Riordan array is an infinite lower triangular matrix, , constructed from two formal power series, () of order 0 and () of order 1, such that , = [] ().. A Riordan array is an element of the Riordan group. [1]
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication and matrix decomposition). The product sometimes includes a permutation matrix as well.
If and are rings and is a (,)-bimodule, then the triangular matrix ring := [] consists of 2-by-2 matrices of the form [], where ,, and , with ordinary matrix addition and matrix multiplication as its operations.
The RQ decomposition transforms a matrix A into the product of an upper triangular matrix R (also known as right-triangular) and an orthogonal matrix Q. The only difference from QR decomposition is the order of these matrices. QR decomposition is Gram–Schmidt orthogonalization of columns of A, started from the first column.
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} } .
At the k-th step (starting with k = 0), we compute the QR decomposition A k = Q k R k where Q k is an orthogonal matrix (i.e., Q T = Q −1) and R k is an upper triangular matrix. We then form A k+1 = R k Q k.
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.