tril

Tril is a matrix operation that returns the lower triangular part of a matrix, including the main diagonal, by zeroing out all elements above the diagonal. In many implementations, an optional offset parameter allows extending or shrinking the included diagonal band.

Mathematically, for a matrix A with entries aij, tril(A) yields a matrix B with bij = aij if i >= j, and bij = 0 otherwise. With an offset k, the condition becomes i >= j - k, so larger k values include more elements above the main diagonal. This makes tril useful for extracting a submatrix that forms a lower triangular region, even for non-square matrices.

Example: for A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], tril(A) is [[1, 0, 0], [4, 5, 0], [7, 8, 9]].

Common implementations and variants:

- MATLAB and Octave: tril(A) and tril(A, k) where k shifts the diagonal band.

- NumPy (Python): numpy.tril(a, k=0) returns the lower triangular portion with an optional offset k.

- R: base lower.tri returns a logical mask; tril can be formed by applying that mask to a matrix if needed.

- Julia: tril(A, k=0) (and related syntax) provides the lower triangular part.

Applications include solving lower-triangular systems, constructing triangular factors in decompositions, and preparing matrices for algorithms that operate on triangular parts, such as forward substitution or sparse matrix manipulations.

See also: triu (the corresponding upper triangular operation), triangular matrices, LU decomposition.