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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 are zero. . Similarly, a square matrix is called upper triangular if all the entries below the main diagonal
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.
One can always write = where V is a real orthogonal matrix, is the transpose of V, and S is a block upper triangular matrix called the real Schur form. The blocks on the diagonal of S are of size 1×1 (in which case they represent real eigenvalues) or 2×2 (in which case they are derived from complex conjugate eigenvalue pairs).
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.
Using these operations, a matrix can always be transformed into an upper triangular matrix (possibly bordered by rows or columns of zeros), and in fact one that is in row echelon form. Once all of the leading coefficients (the leftmost nonzero entry in each row) are 1, and every column containing a leading coefficient has zeros elsewhere, the ...
There is also a real Schur decomposition. If A is an n × n square matrix with real entries, then A can be expressed as [4] = where Q is an orthogonal matrix and H is either upper or lower quasi-triangular. A quasi-triangular matrix is a matrix that when expressed as a block matrix of 2 × 2 and 1 × 1 blocks is
The RQ decomposition transforms a matrix A into the product of an upper triangular matrix R (also known as right-triangular) and an orthogonal matrix Q. The only difference from QR decomposition is the order of these matrices. QR decomposition is Gram–Schmidt orthogonalization of columns of A, started from the first column.
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.