12.3 Tensor Object Addition Operation

Tensor Object Addition Operation is the specific procedure of combining two tensors of identical variance type into a single new tensor of that same type by summing their corresponding components directly, one of the most basic operations available in tensor algebra and the foundation for treating tensors of a fixed type as elements of a vector space.

Foundational Setting

The Prerequisite of Matching Type

Before addition can be applied, the two tensors involved must share an identical variance type $$(p,q)$$ and must be defined over the same underlying vector space, since the operation is defined by pairing components that occupy corresponding index positions.

Component-Wise Definition

Given two tensors $$S$$ and $$T$$ of the same type, their sum $$R$$ is defined entry by entry:

Why the Sum Remains a Tensor

Distributing the Transformation Law

The defining property that must be checked, and holds automatically, is that the sum transforms correctly under a change of basis. Since both $$S$$ and $$T$$ individually satisfy the transformation law with the same matrix $$A$$, their component-wise sum inherits this behavior directly:

Linearity as the Underlying Reason

This works because the tensor transformation law is linear in the components being transformed, so it distributes over addition exactly as ordinary matrix multiplication distributes over the sum of two vectors, guaranteeing that the addition operation never produces an object failing to qualify as a tensor.

Algebraic Properties of the Operation

Commutativity and Associativity

Tensor addition inherits commutativity and associativity directly from the corresponding properties of ordinary numerical addition applied componentwise:

The Zero Tensor and Additive Inverses

A zero tensor, with every component equal to zero in any basis, acts as an identity for addition, and every tensor has an additive inverse obtained by negating every component, satisfying $$T+(-T)=0$$.

Visual Overview

Diagram of Component-Wise Addition

Interaction with Other Operations

Distributing Over Scalar Multiplication

Addition interacts predictably with scalar multiplication, satisfying the standard distributive law:

Distributing Over the Tensor Product

Addition also distributes over the tensor product, meaning the product of a tensor with a sum equals the sum of the individual products:

Summary of Key Traits

Defining Characteristics

• Tensor addition is defined component-wise between two tensors of identical variance type and underlying vector space.
• The sum remains a valid tensor because the transformation law is linear and distributes cleanly over the componentwise sum.
• The operation is commutative and associative, admits a zero tensor as identity, and every tensor has an additive inverse.
• Addition distributes over both scalar multiplication and the tensor product, integrating cleanly with the rest of tensor algebra's operations.