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Other names used for these matrices are unit (lower or upper) triangular, or very rarely normed (lower or upper) triangular. However, a unit triangular matrix is not the same as the unit matrix, and a normed triangular matrix has nothing to do with the notion of matrix norm. All finite unitriangular matrices are unipotent.
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.
where L is a lower unit triangular (unitriangular) matrix, and D is a diagonal matrix. That is, the diagonal elements of L are required to be 1 at the cost of introducing an additional diagonal matrix D in the decomposition. The main advantage is that the LDL decomposition can be computed and used with essentially the same algorithms, but ...
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 =, 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
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. For example, the 5 × ...
An Toeplitz matrix may be defined as a matrix where , =, for constants , …,. The set of n × n {\displaystyle n\times n} Toeplitz matrices is a subspace of the vector space of n × n {\displaystyle n\times n} matrices (under matrix addition and scalar multiplication).
Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices, the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix.