# What does the dot product of a tensor and a vector represent?

The dot product (or inner product) of a tensor T and a vector a produces a vector b = T . a:

$$b_i = T_{ij}a_j = \begin{pmatrix} T_{11} a_1 + T_{12} a_2 + T_{13} a_3\\ T_{21} a_1 + T_{22} a_2 + T_{23} a_3\\ T_{31} a_1 + T_{32} a_2 + T_{33} a_3 \end{pmatrix}$$

The operation is non-commutative if T is non-symmetric, hence for an arbitrary vector $$\mathbf{c}$$, $$\mathbf{c} = \mathbf{a} \cdot \mathbf{T} = \mathbf{T}^\mathrm{T} \cdot \mathbf{a}$$

I have seen the use of this operation in calculations of viscous forces on a body within the computational fluid dynamics context.

The meaning of the dot product of two vectors has been well explained below:

But, what is the meaning of the dot product of a tensor and a vector, if there is any?

• Is this just matrix multiplication? May 9, 2022 at 17:11

## 1 Answer

If you compare your formula for $$T_{ij}a_j$$ to that of matrix multiplication of a matrix $$A$$ with $$ij$$ entry $$A_{ij}$$ and a vector $$v=(v_1,...,v_n)$$, you will see that $$T\cdot a$$ is just matrix multiplication of the matrix $$T$$ with entries $$T_{ij}$$ by the vector $$a=(a_1,...,a_n)$$. So the short answer to your question is 'it represents everything matrix multiplication represents'. To give a longer answer we can try to explain why we get this formula for $$T a$$.

Given a vector space $$V$$, an "$$n$$-tensor" refers to an element of the vector space $$\otimes_{i=1}^n V$$, where $$\otimes$$ is the tensor product. Then a vector in $$v$$ is a 1-tensor, and what you've called $$T$$ is a 2-tensor.

The dot product allows us to identify an element of $$V$$ with a linear map from $$V\to \mathbb{R}$$, because from $$v\in V$$ we can define the linear map $$L:V\to\mathbb{R}$$ by $$L(w) = v\cdot w$$ for any $$w\in V$$. This gives us an operation we can apply on pairs of vectors.

Using this fact we can identify the space of 2-tensors, $$V\otimes V$$ with the space of linear maps $$V\to V$$ by sending a pure 2-tensor $$a\otimes b$$ to the linear map $$L_{ab}$$ taking $$v\to(v\cdot b) a$$ and extending linearly. We get matrices from linear maps by writing the map in a basis, so if we pick a basis $$\{e_i\}_{i=1}^{\dim V}$$ for $$V$$, we get a basis of the pairs $$\{e_i\otimes e_j\}$$ for $$V$$. Then as we learned in our lin alg classes, the $$ij$$th entry of the matrix representing $$T:V\to V$$ in the $$e_i$$ basis is $$T_{ij} = e_i \cdot (Te_j).$$

So in the end we see that the dot product identifies the tensor $$T$$ with a linear map $$V\to V$$, and in a basis we recover the formula for matrix multiplication.