Tensors are mathematical objects that generalize scalars, vectors, and matrices to higher dimensions or ranks. A scalar is a single number and is considered a rank 0 tensor. A vector is a one-dimensional array of numbers and is a rank 1 tensor. A matrix is a two-dimensional array and represents a rank 2 tensor. As we move to higher dimensions, we get tensors of rank 3 and beyond, which are multi-dimensional arrays of numbers.
Tensors have important properties that make them valuable in physics and engineering. They transform predictably between coordinate systems, which means that if we change our reference frame, tensors follow specific transformation rules. This property ensures that physical laws described by tensors remain invariant regardless of the observer's reference frame. For example, a vector, which is a rank-1 tensor, transforms using a rotation matrix when we rotate our coordinate system. Higher-rank tensors follow more complex transformation rules, but the principle remains the same.
Tensors are fundamental in physics for describing various phenomena. The stress tensor, a rank-2 tensor, describes how forces distribute through materials, with components representing different directional stresses. The electromagnetic tensor elegantly unifies electric and magnetic fields into a single mathematical object, showing they're different aspects of the same phenomenon. In Einstein's General Relativity, the metric tensor defines the geometry of spacetime, determining how distances are measured. Other important examples include the inertia tensor in rotational dynamics and the strain tensor that measures material deformation.
In modern machine learning, especially deep learning, tensors are the fundamental data structures. Frameworks like TensorFlow and PyTorch are built around tensor operations. In a neural network, everything is represented as tensors: input data, weight parameters, activations, and gradients. For example, an image input might be a 3D tensor with dimensions for height, width, and color channels. As data flows through a neural network, these tensors undergo transformations through operations like convolutions, matrix multiplications, and non-linear activations. The weight parameters that define these transformations are also stored as tensors. Tensors enable efficient parallel computation on GPUs and support automatic differentiation, which is essential for training neural networks.
To summarize what we've learned: Tensors are mathematical objects that generalize from simple scalars to vectors, matrices, and higher-dimensional arrays. They have the crucial property of transforming predictably between coordinate systems, which makes them ideal for describing physical laws that remain invariant regardless of the observer's reference frame. In physics, tensors elegantly describe complex phenomena like stress in materials, electromagnetic fields, and the geometry of spacetime in relativity. In modern machine learning, tensors serve as the fundamental data structure, representing multi-dimensional data and model parameters while enabling efficient computation on specialized hardware. Understanding tensors provides a powerful mathematical framework for working across disciplines from theoretical physics to artificial intelligence.