uppertriangular

In linear algebra, an n-by-n matrix A is upper triangular if all entries below the main diagonal are zero: a_ij = 0 whenever i > j. The main diagonal can contain any elements. A matrix with zeros on and below the diagonal is strictly upper triangular.

A typical example is the 3×3 matrix with nonzero elements on or above the main diagonal, such as [ [a, b, c], [0, d, e], [0, 0, f] ].

The collection of all n×n upper triangular matrices over a field forms a subspace of the space of all n×n matrices and has dimension n(n+1)/2. The product of two upper triangular matrices is upper triangular, and the inverse of an invertible upper triangular matrix is also upper triangular. Upper triangular matrices are closed under addition, scalar multiplication, and multiplication.

Key properties include: the determinant is the product of the diagonal entries; the trace is the sum of the diagonal entries; the eigenvalues are exactly the diagonal entries (counted with multiplicity). A matrix is invertible if and only if all diagonal entries are nonzero; in that case, its inverse is upper triangular.

Applications include solving linear systems efficiently by back substitution, since the equations proceed from the last row upward. Upper triangular matrices arise in LU decomposition, where a matrix is factored into a lower triangular matrix L and an upper triangular matrix U. Related concepts include lower triangular matrices, diagonal matrices (where only diagonal entries may be nonzero), and unitriangular matrices (upper triangular with ones on the diagonal).