Tensors: Geometry and Applications J.M. Landsberg - Texas A&M ...

2.3. Tensor products 33
to V ⊗W ∗ , by Exercise 2.3.2.(3) below, and as the space of bilinear maps
V × W ∗ → C. If one chooses bases and represents f ∈ V ∗⊗W by a v × w
matrix X = (fi s), the first action is multiplication by a column vector v ↦→
Xv, the second by right multiplication by a row vector β ↦→ βX, the third
by, given an w × v matrix Y = (gs �
i ), taking i,s fi sgs i , and the fourth by
(v,β) ↦→ f i s viβ s .
Exercise 2.3.1.1: Show that the rank one elements in V ⊗W span V ⊗W.
More precisely, given bases (vi) of V and (ws) of W, show that the vw
vectors vi⊗ws provide a basis of V ⊗W.
Exercise 2.3.1.2: Let v1,... ,vn be a basis of V with dual basis α 1 ,...,α n .
Write down an expression for a linear map as a sum of rank one maps
f : V → V such that each vi is an eigenvector with eigenvalue λi, that is
f(vi) = λivi for some λi ∈ C. In particular, write down an expression for
the identity map (case all λi = 1).
Definition 2.3.1.3. Let V1,... ,Vk be vector spaces. A function
(2.3.1) f : V1 × ... × Vk → C
is multilinear if it is linear in each factor Vℓ. The space of such multilinear
functions is denoted V ∗
1 ⊗V ∗
2 ⊗ · · · ⊗ V ∗
k , and called the tensor product of the
vector spaces V ∗
1 ,... ,V ∗
k . Elements T ∈ V ∗
1 ⊗ · · · ⊗ V ∗
k are called tensors.
The integer k is sometimes called the order of T. The sequence of natural
numbers (v1,... ,vk) is sometimes called the dimensions of T.
More generally, a function
(2.3.2) f : V1 × ... × Vk → W
is multilinear if it is linear in each factor Vℓ. The space of such multilinear
functions is denoted V ∗
1 ⊗V ∗
2 ⊗ ...⊗V ∗
k ⊗W, and called the tensor product of
V ∗
1 ,... ,V ∗
k ,W.
If f : V1 × V2 → W is bilinear, define the left kernel
Lker(f) = {v ∈ V1 | f(v1,v2) = 0 ∀v2 ∈ V2}
and similarly for the right kernel Rker(f). For multi-linear maps one analogously
defines the i-th kernel.
When studying tensors in V1⊗ · · · ⊗ Vn, introduce the notation Vˆj :=
) ⊂ Vˆj
→ Vˆj .
V1⊗ · · · ⊗ Vj−1⊗Vj+1⊗ · · · ⊗ Vn . Given T ∈ V1⊗ · · · ⊗ Vn, write T(V ∗
j
for the image of the linear map V ∗
j
Definition 2.3.1.4. Define the multilinear rank (sometimes called the duplex
rank or Tucker rank ) of T ∈ V1⊗ · · · ⊗ Vn to be the n-tuple of natural
numbers Rmultlin(T) := (dim T(V ∗
1 ),... ,dim T(V ∗ n )).