2tensor

2tensor is a term commonly used to denote a second-order tensor, a mathematical object that generalizes scalars and vectors to a multilinear setting. In finite-dimensional vector spaces, a 2tensor can be represented by a matrix of components relative to a chosen basis, making it a central object in linear algebra, physics, and engineering.

Formally, a 2tensor can be viewed as a bilinear map T: V × V → F, or equivalently as a linear map from a vector space to its dual, T: V → V*. Its components Tij with respect to a basis form a two-dimensional array. The matrix representation of T depends on the chosen basis, and tensor transformations describe how these components change under a change of basis.

Two common properties are symmetry and skew-symmetry. A symmetric 2tensor satisfies Tij = Tji, and a skew-symmetric one satisfies Tij = −Tji. The trace of a 2tensor is the sum of its diagonal entries, and contractions reduce a 2tensor to a 1-tensor or a scalar, depending on the indices being contracted.

Applications of 2tensors span physics, engineering, and statistics. The stress tensor and moment of inertia are classic examples in mechanics, while the covariance matrix is a prominent 2tensor in statistics and data analysis. In computer graphics and relativity, second-order tensors describe distortions and projections of space.

In computation, a 2tensor is often stored as a two-dimensional array and manipulated using matrix operations. Matrix multiplication, transposition, and contraction with vectors implement the standard tensor algebra operations. Software libraries such as NumPy and PyTorch support these structures through 2D arrays and related routines.

See also: tensor, second-order tensor, matrix, bilinear form, multilinear algebra.