# Endomorphisms, Tensors in Disguise |18 August 2020|tags: math.LA

Unfortunately the radio silence remained, the good news is that even more cool stuff is on its way*!

For now, we briefly mention a very beautiful interpretation of -tensors. Given some vectorspace , let be the set of linear endomorphisms from to . Indeed, for finite-dimensional vector spaces one can just think of these maps as being parametrized by matrices. Now, we recall that -tensors are multilinear maps which take an element of and its dual space and map it to some real number, that is, .

The claim is that for finite-dimensional the set is isomorphic to the set of -tensors . This is rather intuitive to see, let and , then, we can write and think of as a matrix. This can be formalized, as is done here by showing there is a bijective map . So, we can identify any with some , that is . Similarly, we can fix and identify with a linear map in the dual space, that is . Regardless, both represent the same tensor evaluation, hence which is indeed the standard dual map definition.

Now we use this relation, recall that discrete-time linear systems are nothing more than maps , that is, linear endomorphims, usually on . Using the tensor interpretation we recover what is sometimes called the ‘‘dual system’’ to , namely (suggestive notation indeed). Using more traditional notation for the primal: and dual: system, we recover that . Then, let and for and being a left- and right-eigenvector of , respectively. We see that this implies that , for the eigenvalue corresponding to and the eigenvalue corresponding to . Hence, in general, the eigenspaces of and are generically orthogonal.

In other words, we recover the geometric interpretation of mapping -volumes under versus .

* See this talk by Daniel.