Member-only story

What Are Tensors Exactly? An Intuitive Explanation

Brody Powell

·13.9k Followers· Follow

Published in What Are Tensors Exactly? Hongyu Guo

5 min read

775 View Claps

56 Respond

Save

Listen

What Are Tensors Exactly?

by Hongyu Guo

4.5 out of 5

Language	:	English
File size	:	5601 KB
Text-to-Speech	:	Enabled
Enhanced typesetting	:	Enabled
Print length	:	246 pages
Screen Reader	:	Supported

Tensors are a mathematical concept that has become essential in various fields such as machine learning, physics, and computer graphics. They are used to represent physical quantities that have both magnitude and direction, and can be thought of as generalizations of vectors and matrices.

In this article, we will provide an intuitive explanation of tensors, making them accessible to a wider audience, including those without a strong mathematical background.

What is a Tensor?

A tensor is a multidimensional array that has both magnitude and direction. It can be thought of as a generalization of a vector, which is a one-dimensional array, and a matrix, which is a two-dimensional array.

Tensors are used to represent physical quantities that have more than one component. For example, a vector can be used to represent the velocity of an object, which has both a magnitude and a direction. A tensor can be used to represent the stress on a material, which has both a magnitude and a direction for each of its components.

Types of Tensors

There are many different types of tensors, each with its own unique properties. The most common types of tensors are:

Scalars: Scalars are the simplest type of tensor. They are one-dimensional arrays with only a magnitude. Examples of scalars include temperature and pressure.
Vectors: Vectors are one-dimensional arrays with both a magnitude and a direction. Examples of vectors include velocity and displacement.
Matrices: Matrices are two-dimensional arrays with both magnitude and direction. Examples of matrices include the rotation matrix and the covariance matrix.
Higher-order tensors: Higher-order tensors are arrays with more than two dimensions. Examples of higher-order tensors include the stress tensor and the strain tensor.

Tensor Operations

Tensors can be added, subtracted, multiplied, and divided just like vectors and matrices. However, there are some additional operations that can be performed on tensors that are not possible with vectors and matrices.

One of the most important tensor operations is the dot product. The dot product of two tensors is a scalar that is equal to the sum of the products of the corresponding components of the two tensors.

Another important tensor operation is the cross product. The cross product of two vectors is a vector that is perpendicular to both of the original vectors.

Applications of Tensors

Tensors are used in a wide variety of applications, including:

Machine learning: Tensors are used to represent the input and output data for machine learning models. They are also used to store the weights and biases of the model.
Physics: Tensors are used to represent the physical properties of materials, such as stress, strain, and elasticity.
Computer graphics: Tensors are used to represent the transformation of objects in 3D space. They are also used to store the lighting and shading information for a scene.

Tensors are a powerful mathematical concept that has a wide range of applications. They can be used to represent physical quantities that have both magnitude and direction, and can be thought of as generalizations of vectors and matrices.

In this article, we have provided an intuitive explanation of tensors, making them accessible to a wider audience, including those without a strong mathematical background.

What Are Tensors Exactly?

by Hongyu Guo

4.5 out of 5