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 (1,1)-tensors. Given some vectorspace mathcal{V}, let mathrm{End}(mathcal{V}) be the set of linear endomorphisms from mathcal{V} to mathcal{V}. Indeed, for finite-dimensional vector spaces one can just think of these maps as being parametrized by matrices. Now, we recall that (1,1)-tensors are multilinear maps which take an element of mathcal{V} and its dual space mathcal{V}^{star} and map it to some real number, that is, T^1_1:mathcal{V}times mathcal{V}^{star}to mathbf{R}.

The claim is that for finite-dimensional mathcal{V} the set mathrm{End}(mathcal{V}) is isomorphic to the set of (1,1)-tensors mathcal{T}_1^1(mathcal{V}). This is rather intuitive to see, let vin mathcal{V} and v^{star}in mathcal{V}^{star}, then, we can write langle v^{star},T_1^1 vrangle and think of T_1^1in mathcal{T}^1_1(mathcal{V}) as a matrix. This can be formalized, as is done here by showing there is a bijective map phi:mathrm{End}(V)to mathcal{T}^1_1(mathcal{V}). So, we can identify any T_1^1 with some Ain mathbf{R}^{ntimes n}, that is T_1^1(v,v^{star})=v^{star}(Av)=langle v^{star},Av rangle. Similarly, we can fix v^{star} and identify T_1^1 with a linear map in the dual space, that is T_1^1(v^{star})(v)=(A^{star}v^{star})(v)=langle A^{star}v^{star},vrangle. Regardless, both represent the same tensor evaluation, hence

 langle v^{star},Av rangle = langle A^{star}v^{star},v rangle,

which is indeed the standard dual map definition.

Now we use this relation, recall that discrete-time linear systems are nothing more than maps vmapsto Av, that is, linear endomorphims, usually on mathbf{R}^n. Using the tensor interpretation we recover what is sometimes called the ‘‘dual system’’ to vmapsto Av, namely v^{star}mapsto A^{top}v^{star} (suggestive notation indeed). Using more traditional notation for the primal: x_{k+1}=Ax_k and dual: z_{k+1}=A^{top}z_k system, we recover that langle z_{k+1},x_k rangle = langle z_k, x_{k+1}rangle forall kin mathbf{Z}. Then, let z_0=w and x_0=v for w_j and v_i being a left- and right-eigenvector of A, respectively. We see that this implies that mu_j^k langle w_j, v_i rangle = lambda_i^k langle w_j, v_i rangle, for mu_j the eigenvalue corresponding to w_j and lambda_i the eigenvalue corresponding to v_i. Hence, in general, the eigenspaces of A and A^{top} are generically orthogonal.

In other words, we recover the geometric interpretation of mapping n-volumes under A versus A^{top}.

* See this talk by Daniel.