unittriangulaire

Unitriangular, or unitriangulaire in French, refers to a class of square matrices that are triangular and have ones on the diagonal. In common usage, it typically denotes upper unitriangular matrices: upper triangular matrices with all diagonal entries equal to 1. This makes the matrix equal to the identity plus a strictly upper triangular part.

An n by n unitriangular matrix U over a field F has the form U = I + N, where I is the identity matrix and N is strictly upper triangular (N has zeros on and below the diagonal). Equivalently, a_{ii} = 1 for all i, a_{ij} ∈ F for i < j, and a_{ij} = 0 for i > j. For example, a 3 by 3 upper unitriangular matrix looks like

[1 a b; 0 1 c; 0 0 1]

with a, b, c ∈ F.

Key properties include that the determinant of a unitriangular matrix is 1, it is invertible, and its inverse is also unitriangular (since N is nilpotent, I + N has inverse I − N + N^2 − …, terminating after a finite number of terms). The set UT(n, F) of all upper unitriangular n×n matrices forms a group under multiplication, known as the unitriangular group. This group is unipotent and nilpotent, with dimension n(n−1)/2 as a manifold over F.

Unitriangular matrices arise in linear algebra and algebraic groups as a standard unipotent subgroup of GL_n(F). They are generated by elementary matrices that add a multiple of one row to another above the diagonal. Variants include lower unitriangular matrices, where ones sit on the diagonal and the nonzero entries lie below it.