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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]
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.
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.
There is also a real Schur decomposition. If A is an n × n square matrix with real entries, then A can be expressed as [4] = where Q is an orthogonal matrix and H is either upper or lower quasi-triangular. A quasi-triangular matrix is a matrix that when expressed as a block matrix of 2 × 2 and 1 × 1 blocks is
In linear algebra, a Block LU decomposition is a matrix decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U. This decomposition is used in numerical analysis to reduce the complexity of the block matrix formula.
Triangular arrays may list mathematical values other than numbers; for instance the Bell polynomials form a triangular array in which each array entry is a polynomial. [ 10 ] Arrays in which the length of each row grows as a linear function of the row number (rather than being equal to the row number) have also been considered.
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.