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...

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...

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

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

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...

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 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...

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

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...

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...

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...

elementary reflectors. It is also referred to as a triangular factor, and is a triangular matrix and they are used in the Householder transformation...

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{...

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...

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

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

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 ,...

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...

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...

vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix). The vector projection of...

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...

matrix. Typical examples of matrices that can take advantage of packed storage include: symmetric or hermitian matrix Triangular matrix Banded matrix...

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...

perturbation theory also leads to matrix eigenvalue problem for degenerate states. Defective matrix Scaling (geometry) Triangular matrix Semisimple operator Diagonalizable...

definite matrix A is A = LL* where L is a lower triangular matrix. An incomplete Cholesky factorization is given by a sparse lower triangular matrix K that...

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 (possibly bordered by rows or columns of zeros)...