Tensor

What is a Tensor?

In mathematics, a tensor is a multidimensional array with a uniform type. It can also be seen as representing a system of equations.

There are several tensor-notation conventions, such as Ricci calculus, Einstein summation convention, and Penrose graphical notation. In general, the tensor notation is very similar to matrix notation, with a capital letter representing a tensor and lowercase letters with subscript integers representing scalar values within the tensor. An example is Einstein tensor used in the relativity field equations and usually denoted as Gμν.

Its algebra is an extension of matrix algebra, and a matrix can be considered as a two-dimensional tensor. Tensor’s applications are a result of its algebraic functionalities, which include reshaping, reduction, and element-wise operations.

This construct is very useful in many areas of physics, such as quantum mechanics, electromagnetic theory, fluid dynamics, elasticity, and relativity theory.

Many of the concepts behind a tensor were outlined by German mathematician Karl Friedrich Gauss in the ninetieth century. In 1846, the Irish mathematician William Rowan Hamilton introduced the term “tensor”. In the twentieth century, tensors became well-known through Albert Einstein’s general relativity theory.

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Figure 1: Einstein Field Equations

 

Why are tensors important?

Tensors are widely used in science and engineering. In physics, their main applications come from their transformation properties, which enable the description of the same laws of physics in all reference systems.

They are also applied in artificial intelligence. For example, in machine learning, they are used in the training and operation of deep learning models. The well-known application TensorFlow derives its name from this mathematical concept.

Tensors + LogicPlum

Tensors are quite difficult to handle, and using them requires advanced mathematical knowledge. They are used by LogicPlum’s platform in deep learning operations. However, as this platform handles them in an automated way, tensor management doesn’t present a problem for its users.

 

Additional Resources

For those wanting to know more about tensors:

Rowland, Todd and Weisstein, Eric W. Tensor. From MathWorld–A Wolfram Web Resource. Available at https://mathworld.wolfram.com/Tensor.html

Carter, T. (2019) What is a tensor and why should I care? Available at https://militaryembedded.com/ai/deep-learning/what-is-a-tensor-and-why-should-i-care

For those wanting to know more about TensorFlow:

TensorFlow’s official website: https://www.tensorflow.org/