Triangular matrix information

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...

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...

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...

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

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

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...

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{...

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...

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

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...

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...

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

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...

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...

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

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...

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

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...

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...

square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the...

{\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...