Generalization of the matrix exponential

Question

I've seen this post which addresses the question of exponentiating a vector. I was wondering if there's a well-defined notion of exponentiating a rank $r$ tensor? For instance, if I have a rank 3 tensor $A_{ijk}$, can I compute something like $\mathrm{Exp}[A_{ijk}]$? If you know of any references/insights on this I'd highly appreciate it!

Answer 1

(Partial answer)

Since exponentiation is generally defined via the series of the same name, i.e. $$e^x = \sum_{k=0}^\infty \frac{x^k}{k!},$$ the object $x$ needs to belong to a space where addition, multiplication and scalar multiplication (due to the factorial prefactor) are themselves defined.

Given that tensors form vector spaces, addition and scalar multiplication are natural operations. Now, you need to define a multiplicative operation between tensors (in such a way that they now form an algebra).

The most natural way to do so corresponds to the case where multiplication is interpreted as the tensorial product. If you want to deal with antisymmetric tensors only, then you may consider the exterior product instead. If they form a Lie algebra, then the Lie brackets will play the role of the multiplicative operation. But nothing prevents you to consider more "exotic" products.

You can also "import" the product from another space through isomorphism, as it is traditionally done for vectors for instance, which are re-interpreted as vectors of the tangent space; for example, one has $\mathbf{v} = v_1\mathbf{e}_1 + v_2\mathbf{e}_2$ $\longleftrightarrow$ $\mathbf{v} \cdot \nabla = v_1\partial_x + v_2\partial_y$, hence $e^{\mathbf{v}} = e^{v_1\partial_x + v_2\partial_y}$, so that multiplication is now seen as operator composition.

So, in the end, you need a tensorial algebra where (tensorial) multiplication is closed, associative (at least power associative) and unital (otherwise the first term of the exponential series wouldn't be defined).