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The transpose of an upper triangular matrix is a lower triangular matrix and vice versa. A matrix which is both symmetric and triangular is diagonal. In a similar vein, a matrix which is both normal (meaning A * A = AA *, where A * is the conjugate transpose) and triangular is also diagonal. This can be seen by looking at the diagonal entries ...
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix multiplication and matrix decomposition). The product sometimes includes a permutation matrix as well.
Related: the LDU decomposition is =, where L is lower triangular with ones on the diagonal, U is upper triangular with ones on the diagonal, and D is a diagonal matrix. Related: the LUP decomposition is P A = L U {\displaystyle PA=LU} , where L is lower triangular , U is upper triangular , and P is a permutation matrix .
The Crout matrix decomposition algorithm differs slightly from the Doolittle method. Doolittle's method returns a unit lower triangular matrix and an upper triangular matrix, while the Crout method returns a lower triangular matrix and a unit upper triangular matrix. So, if a matrix decomposition of a matrix A is such that: A = LDU
of a permutation matrix (which has exactly a single in each column, and otherwise zeros), a lower triangular matrix and an upper triangular matrix . The determinants of the two triangular matrices L {\displaystyle L} and U {\displaystyle U} can be quickly calculated, since they are the products of the respective diagonal entries.
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.
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's 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 .