Triangular matrix

A square matrix whose entries are all zero either above or below its main diagonal. It is upper triangular when entries below the diagonal vanish, and lower triangular when entries above it vanish.

Gaussian elimination: A method for solving linear systems by applying row operations to reduce a matrix to echelon form. Elimination transforms many systems into triangular form, enabling efficient back-substitution.
Square matrix: A matrix with the same number of rows and columns. Triangularity is defined only for square matrices.
Symmetric matrix: A square matrix equal to its transpose. Symmetry reflects entries across the diagonal rather than requiring one triangular half to vanish.
Tridiagonal matrix: A matrix whose nonzero entries lie on the main diagonal and the diagonals immediately above and below it. It is a sparse structure that is not generally triangular but often admits efficient solution methods.
Back substitution: A procedure for solving a triangular linear system by finding variables successively from the last equation upward. Upper triangular systems can be solved directly by this procedure.
Main diagonal: The entries of a matrix whose row and column indices are equal. The diagonal separates the entries required to vanish from those that may remain.
Hermitian matrix: A complex square matrix equal to its conjugate transpose. Hermitian structure relates opposite sides of the diagonal, unlike triangularity’s zero region.
Cholesky decomposition: A factorization of a positive-definite Hermitian matrix into a triangular factor and its conjugate transpose. Its triangular factor turns positive-definite linear systems into two triangular solves.
Determinant: A scalar associated with a square matrix that encodes properties including invertibility and signed volume scaling. A triangular matrix’s determinant is the product of its diagonal entries.
Upper triangular matrix: A square matrix whose entries below the main diagonal are zero. This is the triangular form with all entries below the diagonal eliminated.

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Gauss–Seidel methodRelated: Each sweep solves a triangular system, making the sequential update order explicit.