In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal...

called an upper triangular matrix. Similarly, if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries...

} Proposition — A normal triangular matrix is diagonal. Proof Let A be any normal upper triangular matrix. Since ( A ∗ A ) i i = ( A A ∗ ) i...

decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U. The systems L ( U x ) = b {\displaystyle...

diagonal matrix. If all entries below (resp above) the main diagonal are zero, A {\displaystyle A} is called an upper (resp lower) triangular matrix. The...

or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product...

the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and,...

factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used...

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...

positive definiteness of a symmetric real matrix. Namely, the matrix is reduced to an upper triangular matrix by using elementary row operations, as in...

of a triangular matrix are precisely its eigenvalues (see Triangular matrix#Properties). An n × n matrix A is diagonalizable if there is a matrix V and...

Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal...

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A  is symmetric ⟺ A = A T . {\displaystyle A{\text{...

and a matrix in row echelon form is (weakly) upper triangular. Thus, the row echelon form can be viewed as a generalization of upper triangular form for...

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 ,...

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...

algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries...

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

to another row. Using these operations, a matrix can always be transformed into an upper triangular matrix, and in fact one that is in row echelon form...

decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for...

a pentadiagonal matrix and so on. Triangular matrices For k1 = 0, k2 = n−1, one obtains the definition of an upper triangular matrix similarly, for k1...

possible for a real or complex matrix to have all real eigenvalues without being Hermitian. For example, a real triangular matrix has its eigenvalues along...

triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example...

perform a QR decomposition, writing the matrix as a product of an orthogonal matrix and an upper triangular matrix, multiply the factors in the reverse order...

{\displaystyle T} is a triangular matrix. For a matrix A {\displaystyle A} of rank r {\displaystyle r} , the triangular matrix T {\displaystyle T} can...

triangular Toeplitz matrices and C {\displaystyle C} is a strictly lower triangular matrix. The convolution operation can be constructed as a matrix multiplication...

from above) yield an upper triangular matrix in order to compute the QR decomposition. In order to form the desired matrix, zeroing elements (2,1) and...

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

_{i}{\big (}|a_{ii}|{\big )}}}} recalling that the eigenvalues of any triangular matrix are simply the diagonal entries. The condition number computed with...

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...