tensor

A tensor is a mathematical object that generalizes scalars, vectors, and matrices. It can be defined as a multilinear map between vector spaces or, equivalently, as an array of components that transform according to a prescribed rule under a change of basis, preserving its geometric meaning.

The rank or order of a tensor indicates the number of indices required to describe its components. A scalar has rank 0, a vector rank 1, and a matrix rank 2; higher ranks describe more complex multi-linear relationships. The components depend on the chosen basis, but the tensor itself is invariant under basis transformations, thanks to its transformation laws. Covariant, contravariant, and mixed indices reflect different ways components transform.

Key operations include the tensor product, which combines two tensors into a higher-rank tensor; contraction, which sums over a pair of opposite indices to reduce rank; and symmetry properties, such as symmetry or antisymmetry in indices. The metric tensor provides a means to raise and lower indices, linking contravariant and covariant components.

In physics and differential geometry, tensors encode physical quantities and geometric relations, including the metric tensor, stress-energy tensor, curvature tensors, and more. In computer science and machine learning, the term tensor is also used for multi-dimensional arrays of data, with tensor arithmetic and automatic differentiation implemented in modern frameworks to run efficiently on hardware accelerators.

The tensor concept originated in the 19th century with the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita; the term was adopted to describe these multilinear objects and is now central across mathematics, physics, and data science.