Review of Multilinear Algebra

by Min Ru

1 Review of Multilinear Algebra

We mainly deal in the following with finite-dimensional vector spaces. Let V be an m-

m dimensional real vector space. If we choose a basis {ei}i=1, V is isomorphic to the Euclidean space Rm by assigining its components to eact element of V . Now we review briefly some methods by which produce new vector spaces out of the given spaces.

• Dual space. V ∗ := {α : V → R; α is a linear map} has the structure of an m-

i dimensional vector space and is called the duea space of V . We define e (ej) := δij, i m ∗ m then {e }i=1 forms a basis of V which is called the dual basis of {ei}i=1.

• Tensor product. Let V and W be vector spaces of dimension m and n. Then the space Hom(V,W ) := {φ : V → W ; φ is a linear map} has the structure of an mn- dimensional vector space. Hom(V ∗,W ), also denoted by V ⊗ W , is called the tensor product of V and W .

For v ∈ V, w ∈ W we define v ⊗ w ∈ V ⊗ W by v ⊗ w(v∗) := v∗(v)w. Then any element of V ⊗ W may be expressed by a linear combination of elements of the form v ⊗ w,

and, in fact, {ei ⊗ fj}1≤i≤m,1≤j≤n forms a basis of V ⊗ W .

It is important to note that V ⊗ W is isomorphic to the vector space

{φ : V ∗ × W ∗ → R; φ is a bilinear map}

by assigning to v ⊗ w the bilinear map: (v∗, w∗) 7→ w∗(v ⊗ w(v∗)) = v∗(v)w∗(w) ∈ R.

1 • Tensor space. For a vector space V , we define the tensor space of type (r, s) of V , which is denoted by

r ∗ ∗ Ts (V ) = V ⊗ · · · ⊗ V ⊗ V ⊗ · · · ⊗ V ,

as the vector space

{φ : V ∗ × · · · × V ∗ × V × · · · × V → R; φ is a multilinear map},

where the first products for V is taken r times, and the second products for V ∗ is taken s times. Its element is called tensors of type (r, s).

∗ ∗ If xi ∈ V (1 ≤ i ≤ r), yj ∈ V (1 ≤ j ≤ rs) are given, then we get an (r, s)-tensor by the following formula:

∗ ∗ ∗ ∗ Y ∗ ∗ x1 ⊗ · · · ⊗ xr ⊗ y1 ⊗ · · · ⊗ ys (x1, . . . , xr, y1, . . . , ys) := xi (xi)yj (yj). i,j

j1 js r Then, we easily see that {ei1 ⊗ · · · ⊗ eir ⊗ e ⊗ · · · ⊗ e } forms a basis for Ts (V ) and r r+s r dim Ts (V ) = m . Thus, every t ∈ Ts (V ) may be expressed as

X i1,...,ir j1 js t = tj1...,js ei1 ⊗ · · · ⊗ eir ⊗ e ⊗ · · · ⊗ e . i1,...,ir,j1...,js

r ∗ ∼ r ∗ r+r0 ∼ We note that we have canonical isomorphisms Ts (V ) = Ts (V ) and Ts+s0 (V ) = r r0 Ts (V ) ⊗ Ts0 (V ).

• Exterior algebra. We recall that the vector space

k ^(V ) := {α : V × · · · V → R; α is a skew-symmetric k-linear map},

where the product is taken k-times. Vk(V ) is called the k-th exterior product of V ∗ and its elements are called the k-forms. Here α is said to be skew-symmetric if for any

2 permutation σ of {1, . . . , k} we have α(xσ(1), ··· , xσ(k)) = sgnσ · α(x1, ··· , xk), where sgnσ denotes the sign of a permutation σ.

For instance, we define

∗ ∗ ∗ x1 ∧ · · · ∧ xk(x1, . . . , xk) := det(xi (xj)).

∗ ∗ Vk ∗ ∗ Then we easily check that x1 ∧ · · · ∧ xr ∈ (V ) and that xσ(1) ∧ · · · ∧ x σ(k) =

∗ ∗ i1 ik Vk sgnσx1 ∧ · · · ∧ xk. Then {e ∧ · · · ∧ e , i1 < ··· < ik} forms a basis for (V ) and Vk (m+k)! V0 V1 ∗ Vk dim (V ) = m!k! . In particular, we have (V ) = R, (V ) = V , (V ) = {0} if k > m. Further we define for α ∈ Vk(V ) and β ∈ Vl(V ) their exterior product α ∧ β ∈ Vk+l(V ) by

1 X α ∧ β(x1, . . . , xk+l) := (sgnσ)α(xσ(1), . . . , xσ(k))β(xσ(k+1), . . . , xσ(k+l)). k!l! σ

kl V∗ Lm Vk Note that α ∧ β = (−1) β ∧ α and (V ) := k=0 (V ) has the structure of an algebra with respect to “∧”.

Now in the same manner, we may construct

^(V ) := {ξ : V ∗ × · · · V ∗ → R; ξ is a skew-symmetric k-linear map}, k

where the product is taken k-times. Then {ei1 ∧ · · · ∧ eik , i1 < ··· < ik} forms a basis V V V for k(V ), and we may considet the exterior product ξ ∧ η ∈ k+l(V ) of ξ ∈ k(V ) V and η ∈ l(V ) as above.

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