In mathematics, particularly in the field of linear algebra and functional analysis, the trace operator is a function that assigns a single number to a square matrix (or more generally, to a linear operator). The trace of a matrix is defined as the sum of its diagonal elements.
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On a bounded Lipschitz domain, the Sobolev trace operator is the continuous linear extension to of the boundary restriction of smooth functions. Continuity and density make this extension unique; its existence is the Sobolev trace theorem. The kernel prescribes a homogeneous Dirichlet boundary condition. Vanishing Sobolev trace on only part of the boundary instead gives the form domain for a mixed boundary condition, with the natural Neumann boundary condition on the remainder. On an assembly of congruent tiles, taking a Sobolev trace commutes with a constant matrix acting on the tile restrictions. This lets a transplantation theorem be proved on quadratic form domains even when corner eigenfunctions have singular classical derivatives.