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...

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

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...

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

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...

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...

In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising...

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...

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...

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

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...

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 ,...

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

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

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...

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{...

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

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...

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

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

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...

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

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...

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...

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...

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...

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)...

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

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