Multilinear Algebra

Honor Diﬀerential Geometry, MATH 5540H, Sp’2014

Instructor: Sergei Chmutov

Handout 4: Multilinear algebra.

Let V be a vector space over R of ﬁnite dimension. Dual vector space V ∗ is a vector space of linear function f : V → R. It has the same dimension as V , so it is isomorphic to V , but not NOT CANONICALLY. Any bilinear form B(·, ·): V × V → R determines a linear map B : V → V ∗ by

(B(u))(v) := B(u, v) for any u, v ∈ V . ∈ | {z } V ∗ Backward any linear map B : V → V ∗ determines a bilinear form B(·, ·) : V × V → R by the same formula. Thus the vector space of bilinear forms coincides with the vector space of linear maps from V to its dual. This vector space is the tensor square of V ∗, V ∗ ⊗ V ∗ =(V ∗)⊗2.

Tensor product. Let V and W be two vector spaces (over R) of dimensions n and m respectively. Consider (infinite dimensional) vector space M spanned by all pairs (v, w) for v ∈ V and w ∈ W . Its elements are finite linear combinations of pairs (v, w). Consider the subspace R ⊂ M spanned by the following linear combinations (v1, w)+(v2, w) − (v1 + v2, w) , (v, w1)+(v, w2) − (v, w1 + w2) , c · (v, w) − (c · v, w) , c · (v, w) − (v,c · w) , for all v ∈ V , w ∈ W , and c ∈ R. The tensor product is defined as the quotient space

V ⊗ W := M/R .

A class of pair (v, w) is denoted by v ⊗ w. Thus the tensor product of vectors satisﬁes the the relations

(v1 + v2) ⊗ w = v1 ⊗ w + v2 ⊗ w , v ⊗ (w1 + w2)= v ⊗ w1 + v ⊗ w2 ,

(c · v) ⊗ w = v ⊗ (c · w)= c · v ⊗ w . ′ ′ Consequently, if {e1,..., en} is a basis for V and {e1,..., em} is a basis for W , then any element of ′ V ⊗ W can be expressed as a linear combinations of ei ⊗ ej. Thus V ⊗ W is of dimension nm with a ′ basis {ei ⊗ ej} for i =1,...,n and j =1,...,m. There are canonical isomorphisms ∼ ∼ ∗ ∼ ∗ ∗ V ⊗ W = W ⊗ V, (V1 ⊗ V2) ⊗ V3 = V1 ⊗ (V2 ⊗ V3), (V ⊗ W ) = V ⊗ W .

Also there is a (universal) bilinear map B : V × W → V ⊗ W, (v, w) 7→ v ⊗ w. It is universal in the following sense. For any vector space U and a biliner map B(·, ·): V × W → U, there is a unique linear

1 map b : V ⊗ W → U such that B(v, w)= b ◦ B(v, w) for any v ∈ V and w ∈ W . In other words, B(·, ·) can be descented to b in the commutative diagram

B(·,·) / V × W o7/ U ooo 7 ooo B oo ooo o b V ⊗ W

In particular, for any bilinear form B(·, ·): V × V → R, there is a linear function b : V ⊗ V → R such that B(u, v) = b ◦ B(u, v). As a linear function, b ∈ (V ⊗ V )∗ ∼= V ∗ ⊗ V ∗. Thus the space of bilinear forms is isomorphic (canonically) to V ∗ ⊗ V ∗. Tensor algebra. A tensor t of type (p,q) on a vector space V is an element of the space

t ∈ V ⊗ V ⊗ ... ⊗ V ⊗ V ∗ ⊗ V ∗ ⊗ ... ⊗ V ∗ = V ⊗p ⊗ (V ∗)⊗q . | p times{z } | q times{z }

1 n ∗ j Endowing V with a basis e1,..., en yields a dual basis e ,..., e for V consisting of linear functions e j j1 jq on V deﬁned by e (ei) := δij. This gives a basis ei1 ⊗ ... ⊗ eip ⊗ e ⊗ ... ⊗ e in the space of tensors V ⊗p ⊗ (V ∗)⊗q. So every tensor can be expressed as a linear combination

i1...ip j1 jq t = X tj1...jq ei1 ⊗ ... ⊗ eip ⊗ e ⊗ ... ⊗ e .

i1...ip The coeﬃcients tj1...jq are called coordinates of the tensor t relative to the basis e1,..., en. Examples.

1. Scalars of R are tensors of type (0, 0) if we set V ⊗0 := R and (V ∗)⊗0 := R.

2. Vectors are tensors of type (1, 0).

3. Covectors are tensors of type (0, 1).

4. Bilinear forms are tensors of type (0, 2). The tensor coordinates of a bilinear form are the entries of its matrix relative to the basis e1,..., en.

j 5. Linear operators A : V → V are tensors of type (1, 1). An elementary tensor ei ⊗ e corresponds to an operator which transforms ej into ei and sends all other basic vectors to zero. More general, the tensor product u ⊗ f of a vector u ∈ V and covector f ∈ V ∗ corresponds to an operator whose image is spanned by u and which acts on v ∈ V as follows: v 7→ f(v)u. In full generality, if a i i linear operator A has a matrix (aj) relative to a basis e1,..., en, that is A(ej) = P ajei. Then, n i j viewing as a (1, 1)-tensor, it can be written as A = X ajei ⊗ e . i,j=1