Tensor

A tensor is a mathematical object that generalizes scalars, vectors, and matrices to arbitrary dimensions. Tensors are fundamental in describing physical phenomena, particularly in mechanics, electromagnetism, and relativity.

Tensors encode relationships between different directions and magnitudes in space. A scalar (rank-0 tensor) has no direction; a vector (rank-1) has one; a matrix (rank-2) has two. Higher-rank tensors extend this pattern, useful for describing complex physical quantities like stress, strain, and curvature.

In mathematical analysis, tensors live in multilinear spaces, where they transform predictably under coordinate changes—a property making them indispensable across disciplines. Engineers use tensors to model material properties. Physicists use them to formulate the laws of general relativity. In machine learning, high-dimensional tensors are the building blocks of neural networks and embeddings.

The beauty of tensors lies in their coordinate-independence: a tensor's intrinsic meaning remains unchanged regardless of which perspective you view it from, making them universally applicable across different representations and computational systems.

Related

Vector space, Linear algebra, Multilinear algebra, Curvature, Stress (physics), Neural networks